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-   -   About the Problem 3.17b (http://book.caltech.edu/bookforum/showthread.php?t=2097)

rpistu 10-12-2012 09:41 AM

About the Problem 3.17b
 
I donít quite understand the Problem 3.17b. What the meaning of minimize E1 over all possible (∆u, ∆v). Instead, I think it should minimize E(u+∆u,v+∆v), starting from the point (u,v)=(0,0). Is the optimal column vector [∆u,∆v]T is corresponding to the vt in the gradient descent algorithm (here, as the problem said, it is -∆E(u,v)), the norm ||(∆u,∆v)||=0.5 corresponding to the step size ɧ, and (u,v) corresponding to the weight vector w? Then, what the meaning of compute the optimal (∆u, ∆v)?

magdon 10-12-2012 02:26 PM

Re: About the Problem 3.17b
 
\hat E_1 is a function of \Delta u,\Delta v. You want to choose \Delta u,\Delta v (the vector to move in) to minimize \hat E_1. The negative gradient direction is going to be the direction to move (this is shown in the chapter) and you have to rescale that so the step size is 0.5.

Quote:

Originally Posted by rpistu (Post 6313)
I don’t quite understand the Problem 3.17b. What the meaning of minimize E1 over all possible (∆u, ∆v). Instead, I think it should minimize E(u+∆u,v+∆v), starting from the point (u,v)=(0,0). Is the optimal column vector [∆u,∆v]T is corresponding to the vt in the gradient descent algorithm (here, as the problem said, it is -∆E(u,v)), the norm ||(∆u,∆v)||=0.5 corresponding to the step size ɧ, and (u,v) corresponding to the weight vector w? Then, what the meaning of compute the optimal (∆u, ∆v)?


rpistu 10-13-2012 08:17 PM

Re: About the Problem 3.17b
 
Yes, E1 is a function of ∆u, ∆v, but it is also a function of u, v. Then, what is the u, v in this function? Still use (0, 0) as part (a) said? Also, what is the ininital value of ∆u, ∆v? In the textbook, it sets w to w(0) at step 0.

Further, does the norm ||(∆u,∆v)||=0.5 means that for each iteration we should ensure that the values of ∆u,∆v meet this resuirements? Another point is that in textbook, we need specify the step size ɧ. However, we could not see any information about the step size.

I don't quite understand the description of the question (Problem 3.17b), so I have so many questions. Could you probably clarify it for me?

Quote:

Originally Posted by magdon (Post 6327)
\hat E_1 is a function of \Delta u,\Delta v. You want to choose math]\Delta u,\Delta v[/math] (the vector to move in) to minimize \hat E_1. The negative gradient direction is going to be the direction to move (this is shown in the chapter) and you have to rescale that so the step size is 0.5.


magdon 10-14-2012 05:43 AM

Re: About the Problem 3.17b
 
Yes, in this problem you can use (u,v)=(0,0) from part (a).

||(∆u,∆v)||=0.5 means that the step size \eta=0.5.

In the chapter we considered two step sizes. First where the step size was fixed at \eta. Second where the step size is proportional to the norm of the gradient. Here, for part (b) the step size is fixed at 0.5.

Quote:

Originally Posted by rpistu (Post 6357)
Yes, E1 is a function of ∆u, ∆v, but it is also a function of u, v. Then, what is the u, v in this function? Still use (0, 0) as part (a) said? Also, what is the ininital value of ∆u, ∆v? In the textbook, it sets w to w(0) at step 0.

Further, does the norm ||(∆u,∆v)||=0.5 means that for each iteration we should ensure that the values of ∆u,∆v meet this resuirements? Another point is that in textbook, we need specify the step size ɧ. However, we could not see any information about the step size.

I don't quite understand the description of the question (Problem 3.17b), so I have so many questions. Could you probably clarify it for me?


rpistu 10-14-2012 04:36 PM

Re: About the Problem 3.17b
 
I have almost understund the problem. But still have a question that what the meaning of the resulting of E(u+∆u,v+∆v) in Part (b), (e-i), and (e-ii). Is it a number or a formula?

Also what the difference of the two parts of (e). One is to minimize E2, the other is to minimize E(u+∆u,v+∆v). So, what the difference of those two?

Quote:

Originally Posted by magdon (Post 6362)
Yes, in this problem you can use (u,v)=(0,0) from part (a).

||(∆u,∆v)||=0.5 means that the step size \eta=0.5.

In the chapter we considered two step sizes. First where the step size was fixed at \eta. Second where the step size is proportional to the norm of the gradient. Here, for part (b) the step size is fixed at 0.5.


luwei0917 10-14-2013 10:40 AM

Re: About the Problem 3.17b
 
I think for part b. we got\begin{bmatrix}\Delta u \\\Delta v\end{bmatrix}
Then use E( u, v) = e^u + e^{2 v} + e^{u v}+ u^2- 3 u v + 4 v^2  - 3 u  - 5 v. Replace u with \Delta u and v with \Delta v we will get a numerical result.


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