Quote:
Originally Posted by virginiatraweek@gmail.com
Let's say that my f function is something simple like y=2x. Let's say that my random points lie on each side of the line such that I end up with the following points:
(1,1,3), (1,3,7), (1,2,3), and (1,4,7). These map ++ and  , since they are on opposite sides of the line.
During the initial step, setting the weights equal to zero yields zero on each of these functions.

Not quite. The "guess" made at each iteration puts these points at zero, but then you compare those zeros with your data, and you see that they should have been 1 or +1 depending on the point. Therefore your current hypothesis is bad, and should be improved, a la w += y*x[n]
Quote:
Originally Posted by virginiatraweek@gmail.com
So, we iterate once by setting the weights equal to 1.

The update step is to move your classifier line by considering *only* 1 misclassified point. And importantly, to update the line in the specific way that makes sure that point will then be classified correctly.
Quote:
Originally Posted by virginiatraweek@gmail.com
Plugging the weights into the first two points yields a positive value. (1+1+3) and (1+3+7) Yet, the bottom two points are still positive. As long as my iterative step is a positive 1, I can't get a negative number in the bottom rows. How does that work? Is that even how the learning is supposed to function?

Take a closer look at the interpretation of w. w is a 3vector that describes a line in 2d. The augmentation we do to make PLA convenient to implement has a specific interpretation of w;
let k =<w[1], w[2]>. k is like a direction of a gradient, where it goes linearly from 0 to 1 in the length of k. w[0] is a distance along k we draw our line at.
This means that the dot product of x[n] and w can be thought of as a measure of "agreement". By the 2d vector analogy from class(which holds in 3d), you can see that in one simple step we can improve this "agreement" from bad to good because we know which way in the 3space must be better (y[n]*x[n]). The miracle of the Perceptron is that just by doing this several times, we *will always* arrive at an acceptable answer.