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Old 10-13-2012, 02:02 PM
erezarnon erezarnon is offline
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Default 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?
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Old 10-13-2012, 02:52 PM
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htlin htlin is offline
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Default Re: Problem 3.7d

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Originally Posted by erezarnon View Post
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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Old 10-13-2012, 07:43 PM
mileschen mileschen is offline
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Default 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?

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Originally Posted by htlin View Post
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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Old 10-14-2012, 05:00 AM
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Default 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).

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Originally Posted by mileschen View Post
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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