Welcome to Gradient Descent! This fundamental optimization algorithm helps us find the minimum of a function. Imagine you're standing on a hill and want to reach the bottom. Gradient descent tells you to look around, find the steepest downward slope, and take a step in that direction. We repeat this process until we reach the valley.
The first step in gradient descent is initialization. We start by choosing initial values for our parameters, often randomly. We also set the learning rate alpha, which determines how big steps we take. A small learning rate means slow but steady progress, while a large learning rate means faster but potentially unstable movement. Finally, we define our cost function that we want to minimize.
Step two is calculating the gradient. The gradient is the derivative of our cost function, which tells us the slope at any point. For our quadratic function, the gradient is simply 2 theta. The gradient points in the direction of steepest increase, like pointing uphill. Since we want to minimize the function, we move in the opposite direction of the gradient, which points downhill toward the minimum.
Step three is updating our parameters using the gradient descent update rule. We subtract the learning rate times the gradient from our current parameter value. In our example, starting at theta equals 2, with learning rate 0.1 and gradient 4, our new parameter becomes 1.6. This moves us closer to the minimum. The learning rate controls how big this step is.
The final step is to repeat the process until convergence. We continue calculating gradients and updating parameters until the algorithm reaches the minimum. Convergence occurs when the gradient approaches zero, meaning we've found the optimal solution. In practice, we stop when changes become very small or after a maximum number of iterations. This iterative process is what makes gradient descent so powerful for optimization problems.