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-   -   Exercises and Problems (http://book.caltech.edu/bookforum/showthread.php?t=258)

rozele 09-28-2012 06:30 PM

Exercise 2.4b
 
The final part of the hint in this question says:
"Now, if you choose the class of these other vectors carefully, then the classification of the dependent vector will be dictated."
The other vectors refers to the set of linearly independent vectors that make up the d+2th vector. What do you mean by class? Do you mean class of vector, (e.g., unit vector), or class based on the PLA algorithm (i.e., +1 or -1)?

magdon 09-29-2012 07:00 AM

Re: Exercise 2.4b
 
Class means \pm1. (Note: there is no PLA or algorithm here; the VC dimension has only to do with the hypothesis set.)

At this point you have established that some input vector x^* is linearly dependent on the others. If you assign the class (\pm1) of the other vectors appropriately, you should be able to show that the linear dependence dictates that the class of x^* must be (say) +1. This means you cannot implement -1 with the other points having those appropriately chosen classifications, and hence this data set cannot be shattered.

This argument will apply to any data set of d+2 points, and so you cannot shatter any set of d+2 points.

Quote:

Originally Posted by rozele (Post 5934)
The final part of the hint in this question says:
"Now, if you choose the class of these other vectors carefully, then the classification of the dependent vector will be dictated."
The other vectors refers to the set of linearly independent vectors that make up the d+2th vector. What do you mean by class? Do you mean class of vector, (e.g., unit vector), or class based on the PLA algorithm (i.e., +1 or -1)?


nahgnaw 09-29-2012 09:44 PM

Problem 2.24
 
When we design the numerical experiment, shall we randomly generate more datasets to determine g_bar(x), E_out, bias, and var?

magdon 09-30-2012 05:16 AM

Re: Problem 2.24
 
Yes, when computing bias and var numerically you need to generate many data sets. For example, \bar g(\mathbf{x}) is the average function that results from learning on each of these data sets.

Quote:

Originally Posted by nahgnaw (Post 5955)
When we design the numerical experiment, shall we randomly generate more datasets to determine g_bar(x), E_out, bias, and var?


RicLouRiv 07-11-2017 05:15 AM

Re: Exercises and Problems
 
Professor -- in my version of the text, for 2.14.b, the inequality is:

2^l > 2Kl^{d_{VC}}.

It looks like others are using a version of the inequality that is:

2^l > Kl^{d_{VC}+1},

which I think makes the problem a little more transparent. I'm wondering if there's a typo in my version? If not, any additional hints on how to treat this version of the inequality would be helpful.


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