Goal: Minimizing the Cost Function

Cost functions (often called loss functions) measure how well a machine learning model is performing. A cost function takes the model's predictions and the actual target values, computing a single number that represents the total error or "cost". A high cost indicates the model's predictions are far off from the actual values, while a low cost shows the predictions are closer to reality.

So, what's the objective when we train a machine learning model? It's typically to make the model as accurate as possible. In the language of cost functions, this translates directly to finding the model parameters that result in the lowest possible value of the cost function.

Think of it like tuning a radio. The parameters of your model (like the slope $m$ and intercept $b$ if you're fitting a line) are the dials you can turn. The cost function tells you how much static (error) you have for a given setting of the dials. Your goal is to adjust those dials until you find the setting that gives you the clearest signal, which means the least amount of static, or the minimum cost.

Why Minimization?

Minimizing the cost function is fundamental because:

1. It Quantifies "Best Fit": The cost function provides a mathematical definition of what "best" means for your model. By minimizing it, you are finding the parameters that make the model fit the training data as well as possible, according to that definition. For instance, minimizing the Mean Squared Error (MSE) means finding the parameters that reduce the average squared difference between predictions and actual values.
2. It Drives Learning: The process of adjusting model parameters to reduce the cost is essentially what "learning" means in many machine learning contexts. The model iteratively changes its parameters, checks the cost, and adjusts again, always trying to move towards a lower cost value.

Consider a very simple scenario where our cost depends on just one parameter, let's call it $w$. The cost function might look something like a bowl shape when plotted against different values of $w$.

A simple parabolic cost function $Cost = w^2$. The goal is to find the value of the parameter $w$ (in this case, $w=0$) that corresponds to the lowest point on the curve, representing the minimum cost.

Our objective is to find the value of $w$ at the very bottom of that bowl. This process of finding the input (or inputs) to a function that result in the minimum output value is called optimization.

In machine learning, we are optimizing the model's parameters to minimize the cost function. While finding the minimum of a simple function like $Cost = w^2$ is straightforward (it's clearly at $w=0$), cost functions for real machine learning models are usually much more complex, depending on potentially millions of parameters.

How do we find this minimum point efficiently, especially when dealing with many parameters? That's where calculus, specifically derivatives, comes into play. As we saw earlier, the derivative tells us the slope or rate of change of a function. We can use this information about the slope to guide us towards the minimum point of the cost function. This is the core idea behind algorithms like gradient descent, which we'll introduce next.