Do we actually take random line in first step of linear regression? 
This is the screenshot I took from a video on linear regression made by Luis Serrano. He explained linear regression step by step (scratch version). The first step was to start with a random line.
The question is do we actually draw a random line, or instead do we perform some calculation like taking an average of y values and initially draw a line. Because if we take any random line it might not fall near any points at all. Maybe it will fall on the 3rd quadrant of the coordinate system where there are no points in this case.

Some methods for robust regression, notably RANSAC (Random sample consensus) are actually built around fitting random lines. But this is, of course, far from what is happening here - I agree with those who say that

*

*it is a pedagogical tool

*the problem can be solved exactly (optimal least squares)

*it is reminiscent of the gradient descent

In the above mentioned robust methods one actually uses exact regression for fitting a line to a random subset of datapoints, thus diminishing the influence of the outliers (to which the exact solution to linear least squares regression is extremely sensitive).
