Gradient descent follows a simple but powerful idea: move in the direction of steepest descent to minimize a function.

The mathematical rule is:

theta = theta - alpha * gradient(theta)

Where:

• theta is the parameter we are optimizing;
• alpha is the learning rate (step size);
• gradient(theta) is the gradient of the function at theta.

1. Define the Function and Its Derivative

We start with a simple quadratic function:

def f(theta):
    return theta**2  # Function we want to minimize

Its derivative (gradient) is:

def gradient(theta):
    return 2 * theta  # Derivative: f'(theta) = 2*theta

• f(theta): this is our function, and we want to find the value of theta that minimizes it;
• gradient(theta): this tells us the slope at any point theta, which we use to determine the update direction.

2. Initialize Gradient Descent Parameters

alpha = 0.3  # Learning rate
theta = 3.0  # Initial starting point
tolerance = 1e-5  # Convergence threshold
max_iterations = 20  # Maximum number of updates

• alpha (learning rate): controls how big each step is;
• theta (initial guess): the starting point for descent;
• tolerance: when the updates become tiny, we stop;
• max_iterations: ensures we don't loop forever.

3. Perform Gradient Descent

for i in range(max_iterations):
    grad = gradient(theta)  # Compute gradient
    new_theta = theta - alpha * grad  # Update rule
    if abs(new_theta - theta) < tolerance:
        print("Converged!")
        break
    theta = new_theta

• Calculate the gradient at theta;
• Update theta using the gradient descent formula;
• Stop when updates are too small (convergence);
• Print each step to monitor progress.

4. Visualizing Gradient Descent

This plots:

• The function curve f(theta) = theta^2;
• Red dots showing gradient descent steps;
• Arrows indicating that the function extends indefinitely.