Whenever you train a model with your data, you are actually producing some new values (predicted) for a specific feature. However, that specific feature already has some values which are real values in the dataset. We know the closer the predicted values to their corresponding real values, the better the model.

Now, we are using cost function to measure how close the predicted values are to their corresponding real values.

We also should consider that the weights of the trained model are responsible for accurately predicting the new values. Imagine that our model is y = 0.9*X + 0.1, the predicted value is nothing but (0.9*X+0.1) for different Xs. [0.9 and 0.1 in the equation are just random values to understand.]

So, by considering Y as real value corresponding to this x, the cost formula is coming to measure how close (0.9*X+0.1) is to Y.

We are responsible for finding the better weight (0.9 and 0.1) for our model to come up with a lowest cost (or closer predicted values to real ones).

Gradient descent is an optimization algorithm (we have some other optimization algorithms) and its responsibility is to find the minimum cost value in the process of trying the model with different weights or indeed, updating the weights.

We first run our model with some initial weights and gradient descent updates our weights and find the cost of our model with those weights in thousands of iterations to find the minimum cost.

One point is that gradient descent is not minimizing the weights, it is just updating them. This algorithm is looking for minimum cost.

It's strange to think about it, but there is more than one measure for how "accurately" a line fits to data points.

To access how accurately a line fits the data, we have a "cost" function which which can compare predicted vs. actual values and provide a "penalty" for how wrong it is.

penalty = cost_funciton(predicted, actual)

A naive cost function might just take the difference between the predicted and actual.

More sophisticated functions will square the value, since we'd rather have many small errors than one large error.

Additionally, each point has a different "sensitivity" to moving the line. Some points react very strongly to movement. Others react less strongly.

Often, you can make a tradeoff, and move TOWARD a point that is sensitive, and AWAY from a point that is NOT sensitive. In that scenario , you get more than you give up.

The "gradient" is a way of measuring how sensitive each point is to moving the line.

This article does a good job of describing WHY there is more than one measure, and WHY some points are more sensitive than others:

https://towardsdatascience.com/wrapping-your-head-around-gradient-descent-with-pictures-3fbd810235f5?source=friends_link&sk=7117e5de8c66bd4a4c2bb2a87a928773