Introduction to Gradient Descent

Finding the minimum point of a cost function is a major goal in training machine learning models. The derivative of a function indicates its slope, or rate of change, at any given point. Setting the derivative to zero ($f'(x) = 0$) can help locate potential minimum points analytically, especially for simple functions.

But what happens when the cost function is complex, perhaps involving millions of parameters (as is common in deep learning)? Solving $f'(x) = 0$ directly might be computationally infeasible or even impossible. We need an iterative approach, a method that can systematically find the minimum without needing to solve complex equations. This is where Gradient Descent comes in.

Think of it like this: imagine you're standing on a foggy mountainside and want to get to the lowest point in the valley. You can't see the entire ground, but you can feel the slope beneath your feet. What's the most straightforward strategy?

1. Check the steepness and direction of the slope where you are currently standing.
2. Take a small step downhill in the steepest direction.
3. Repeat the process: check the slope at your new location and take another step downhill.

By repeating these steps, you'll gradually make your way down towards the valley floor.

Gradient Descent is the mathematical equivalent of this process. Our "position" on the mountainside is the current value of our function's input variable (or variables, as we'll see later). The "slope" we feel is given by the derivative (or its multi-variable equivalent, the gradient). "Taking a step downhill" means adjusting the input variable in the direction opposite to the slope.

Let's consider a simple cost function $J(w)$, where $w$ represents a parameter we want to optimize (like the slope $m$ in a linear model $y=mx+b$).

• We start with an initial guess for $w$.
• We calculate the derivative of the cost function at that point, $J'(w)$. This tells us the slope.
• If $J'(w)$ is positive, the function is increasing at this point. To go "downhill," we need to decrease $w$.
• If $J'(w)$ is negative, the function is decreasing at this point. To go "downhill," we need to increase $w$.

Notice a pattern: we always want to move $w$ in the direction opposite to the sign of the derivative. We can achieve this with a simple update rule:

$$w_{new} = w_{old} - \alpha \cdot J'(w_{old})$$

Here:

• $w_{new}$ is the updated value of our parameter.
• $w_{old}$ is the current value.
• $J'(w_{old})$ is the derivative (slope) calculated at the current value $w_{old}$.
• $\alpha$ (alpha) is a small positive number called the learning rate. It controls how big of a step we take. We'll discuss this more later, but for now, think of it as ensuring our steps are small enough not to overshoot the minimum.

If $J'(w_{old})$ is positive (slope points uphill), subtracting $\alpha \cdot J'(w_{old})$ makes $w_{new}$ smaller than $w_{old}$, moving us left (downhill). If $J'(w_{old})$ is negative (slope points downhill), subtracting a negative value means we add $\alpha \cdot |J'(w_{old})|$, making $w_{new}$ larger than $w_{old}$, moving us right (also downhill). The update rule automatically handles the direction!

We repeat this update step iteratively. With each step, we calculate the new slope at the new position and take another step downhill. As we get closer to the minimum, the slope $J'(w)$ gets closer to zero, meaning the steps become smaller and smaller. We typically stop the process when the steps become very tiny, indicating we've likely converged near the minimum point.

Gradient descent is a foundational optimization algorithm used extensively in machine learning to adjust model parameters and minimize the cost function, thereby improving the model's performance on the training data. It doesn't guarantee finding the absolute global minimum (it might settle in a local minimum, a small dip that isn't the lowest overall point), but in practice, it works remarkably well for many machine learning problems.