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-   -   Problem 3.7d (http://book.caltech.edu/bookforum/showthread.php?t=2120)

erezarnon 10-13-2012 02:02 PM

Problem 3.7d
 
The problem asks us to prove that the optimal column vector is in the opposite direction of the inverse of the Hessian times the gradient.

But didn't the chapter prove that the optimal column vector is in the opposite direction of the gradient?

htlin 10-13-2012 02:52 PM

Re: Problem 3.7d
 
Quote:

Originally Posted by erezarnon (Post 6354)
The problem asks us to prove that the optimal column vector is in the opposite direction of the inverse of the Hessian times the gradient.

But didn't the chapter prove that the optimal column vector is in the opposite direction of the gradient?

The chapter shows that the optimal column vector, subject to the first-order Taylor's approximation, is the negative gradient. Problem 3.7(d) asks you to consider second-order Taylor's approximation instead, though.

Hope this helps. :)

mileschen 10-13-2012 07:43 PM

Re: Problem 3.7d
 
Could you possibly redescribe the Problem 3.17b for me? I don't quite understand the requirements of this question. What's the relation between it and the gradient descent algorithm for logistic regression of the textbook?

Quote:

Originally Posted by htlin (Post 6355)
The chapter shows that the optimal column vector, subject to the first-order Taylor's approximation, is the negative gradient. Problem 3.7(d) asks you to consider second-order Taylor's approximation instead, though.

Hope this helps. :)


magdon 10-14-2012 05:00 AM

Re: Problem 3.7d
 
In gradient descent we studied a similar problem: find the direction to move to minimize the error the most for a given step size. This direction was the negative gradient of E. You can use that fact to solve part (b), because the gradient approximation is exact for linear functions.

Part (a) defined a function \hat E_1. If you set (u,v)=(0,0), \hat E_1 becomes a function of \Delta u,\Delta v. You want to minimize this function under the constraint that \|(\Delta u,\Delta v\|=0.5.

If you choose to use the gradient hint, the gradient of E_1 is related to the coefficients a_u,a_v defined in part (a).

Quote:

Originally Posted by mileschen (Post 6358)
Could you possibly redescribe the Problem 3.17b for me? I don't quite understand the requirements of this question. What's the relation between it and the gradient descent algorithm for logistic regression of the textbook?



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