LFD Book Forum Problem 3.7d
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#1
10-13-2012, 02:02 PM
 erezarnon Junior Member Join Date: Oct 2012 Posts: 3
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?
#2
10-13-2012, 02:52 PM
 htlin NTU Join Date: Aug 2009 Location: Taipei, Taiwan Posts: 601
Re: Problem 3.7d

Quote:
 Originally Posted by erezarnon 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.
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#3
10-13-2012, 07:43 PM
 mileschen Member Join Date: Sep 2012 Posts: 11
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 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.
#4
10-14-2012, 05:00 AM
 magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 596
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 You can use that fact to solve part (b), because the gradient approximation is exact for linear functions.

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

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

Quote:
 Originally Posted by mileschen 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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