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-   -   Intuition of the step of PLA (http://book.caltech.edu/bookforum/showthread.php?t=4614)

henry2015 08-24-2015 06:28 AM

Intuition of the step of PLA
 
According to the book, the update rule for PLA is w(t+1) = w(t) + y(t)x(t), and the book mentions "this rule moves the boundary in the direction of classifying x(t) correctly".

I understand that there is a convergence proof for PLA. But it is hard for me to see why such rule (or step) moves the boundary in the direction of classifying x(t) correctly. The formula just adds actual outcome (i.e. y(t)) times the misclassified point (i.e. x(t)) to the current weight matrix (which is just a vector of coefficient of hypothesis equation).

Any pointer will help.

Thanks in advance!

yaser 08-24-2015 08:34 PM

Re: Intuition of the step of PLA
 
The point {\bf x}_n would be correctly classified if {\bf w}^{\rm T}{\bf x}_n agreed in sign with y_n. Therefore, moving {\bf w}^{\rm T}{\bf x}_n in the direction of agreeing with that sign would be moving it in the right direction.

Adding {\bf x}_n y_n to {\bf w} will indeed achieve that, since it will add {\bf x}_n^{\rm T}{\bf x}_n y_n to the quantity {\bf w}^{\rm T}{\bf x}_n and what it adds agrees with y_n in sign since the {\bf x}_n^{\rm T}{\bf x}_n part is always positive.

elyakim 08-27-2015 02:04 AM

Re: Intuition of the step of PLA
 
Quote:

Originally Posted by yaser (Post 12013)
The point {\bf x}_n would be correctly classified if {\bf w}^{\rm T}{\bf x}_n agreed in sign with y_n. Therefore, moving {\bf w}^{\rm T}{\bf x}_n in the direction of agreeing with that sign would be moving it in the right direction.

Adding {\bf x}_n y_n to {\bf w} will indeed achieve that, since it will add {\bf x}_n^{\rm T}{\bf x}_n y_n to the quantity {\bf w}^{\rm T}{\bf x}_n and what it adds agrees with y_n in sign since the {\bf x}_n^{\rm T}{\bf x}_n part is always positive.

Earlier I indicated having difficulty with reading the equations in problem 1.3.
It works but I'm concerned I'm updating weights with a rule that is "not so smart":
  • the difference between the target function value and x2 (or the y-value in a visual simulation) for a misclassified point.
Especially I don't recognize the product of "x transpose and x" part.

To summarize my questions:
  1. would insight into vector computation make 'everything easier'?
  2. :confused: what is p in the equation? A random symbol?
Thanks again.

henry2015 08-27-2015 06:43 AM

Re: Intuition of the step of PLA
 
Quote:

Originally Posted by yaser (Post 12013)
The point {\bf x}_n would be correctly classified if {\bf w}^{\rm T}{\bf x}_n agreed in sign with y_n. Therefore, moving {\bf w}^{\rm T}{\bf x}_n in the direction of agreeing with that sign would be moving it in the right direction.

Adding {\bf x}_n y_n to {\bf w} will indeed achieve that, since it will add {\bf x}_n^{\rm T}{\bf x}_n y_n to the quantity {\bf w}^{\rm T}{\bf x}_n and what it adds agrees with y_n in sign since the {\bf x}_n^{\rm T}{\bf x}_n part is always positive.

Now, mathematically, I can see why adding y(t) * transpose of x(t) * x(t) to transpose of w(t) * x(t) makes it get closer to the solution. Thanks!

Just a bit hard to visualize it.


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