Intuition of the step of PLA

henry2015 · #1 08-24-2015, 07:28 AM

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 · #2 08-24-2015, 09:34 PM

The point ${\bf x}_n$ would be correctly classified if ${\bf w}^{\rm T}{\bf x}_n$ agreed in sign with

. 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

in sign since the ${\bf x}_n^{\rm T}{\bf x}_n$ part is always positive.

elyakim · #3 08-27-2015, 03:04 AM

Quote:

Originally Posted by yaser

The point ${\bf x}_n$ would be correctly classified if ${\bf w}^{\rm T}{\bf x}_n$ agreed in sign with

. 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

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:

would insight into vector computation make 'everything easier'?
what is p in the equation? A random symbol?

Thanks again.

henry2015 · #4 08-27-2015, 07:43 AM

Quote:

Originally Posted by yaser

The point ${\bf x}_n$ would be correctly classified if ${\bf w}^{\rm T}{\bf x}_n$ agreed in sign with

. 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

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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