Re: Exercises and Problems  1.3
For question 1.3 we are asked to argue that the move from w(t) to w(t+1) is a move "in the right direction". I think I may be misunderstanding the question and/or the figure 1.3. My impression is that the figure shows us the case of R2 (ie, d=1), but that for arbitrary Rd we are considering the case where we change w(t+1) and then argue that the resulting change in the location of the boundary plane is a move toward (x, y) and therefore more likely than not to pass by (x, y), thereby putting (x, y) on the opposite (and correct) side of the classification boundary.
This is easy enough to say in English, as I just did. But unless I'm missing something, I don't think the analytic proof is necessarily so straightforward. It seems to me that I'd have to show that the plane moves closer to the point (x, y). I think then I would have to argue that moving closer implies increasing the likelihood of passing by/through the point. (This part seems straightforward). And then I would argue that increasing the likelihood of passing by/through the point is logically equivalent to increasing the likelihood of correctly classifying the point.
Is my overarching logic here sound? And is a mathematical (specifically, analytic) proof of this argument what you are intending for an answer? Or is the intent just for the reader to formulate an English explanation such as that which I attempt to give above?
Thank you very much for the help! I am really enjoying your book.
