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-   Chapter 2 - Training versus Testing (http://book.caltech.edu/bookforum/forumdisplay.php?f=109)
-   -   Exercises and Problems (http://book.caltech.edu/bookforum/showthread.php?t=258)

yaser 03-25-2012 12:25 AM

Exercises and Problems
 
Please comment on the chapter problems in terms of difficulty, clarity, and time demands. This information will help us and other instructors in choosing problems to assign in our classes.

Also, please comment on the exercises in terms of how useful they are in understanding the material.

tadworthington 07-02-2012 02:36 PM

ERRATA: Small mistake in description of Exercise 2.1
 
I won't "mathify" this correction, as I don't know how in this forum (my LaTex has escaped me after years of neglect!). It's a minor point, but I feel like it should be corrected. In the wording for Exercise 2.1 on page 45:

ERROR: "Verify that m_H(n) < 2^k"
CORRECTION: "Verify that m_H(k) < 2^k"

Thanks!

yaser 07-02-2012 03:58 PM

Re: ERRATA: Small mistake in description of Exercise 2.1
 
Quote:

Originally Posted by tadworthington (Post 3334)
I won't "mathify" this correction, as I don't know how in this forum (my LaTex has escaped me after years of neglect!). It's a minor point, but I feel like it should be corrected. In the wording for Exercise 2.1 on page 45:

ERROR: "Verify that m_H(n) < 2^k"
CORRECTION: "Verify that m_H(k) < 2^k"

Thanks!

Thank you for catching this!

vsthakur 08-24-2012 09:20 AM

Problem 2.9 : Growth function of perceptron, seems incorrect
 
The problem says that in case of perceptron in d-dimensional space, growth function is equal to B(N,k). Consider the following case :

d = 2, implies dvc = 2+1 = 3
N = 4
B(N,k) = 15
but the maximum no. of dichotomies possible in this case is only 14

Can someone please comment if i am missing something.

Thanks.

Vishwajeet.

magdon 08-24-2012 01:17 PM

Re: Problem 2.9 : Growth function of perceptron, seems incorrect
 
Thanks for catching this erratum. The problem shows the upper bound based on the VC dimension. The actual growth function is given by:

2\sum_{i=0}^d \left({N-1}\atop i\right).

Our appologies.

Quote:

Originally Posted by vsthakur (Post 4384)
The problem says that in case of perceptron in d-dimensional space, growth function is equal to B(N,k). Consider the following case :

d = 2, implies dvc = 2+1 = 3
N = 4
B(N,k) = 15
but the maximum no. of dichotomies possible in this case is only 14

Can someone please comment if i am missing something.

Thanks.

Vishwajeet.


vsthakur 08-25-2012 09:04 PM

Re: Problem 2.9 : Growth function of perceptron, seems incorrect
 
Thank you

vsthakur 08-29-2012 02:39 AM

Re: Problem 2.9 : Growth function of perceptron, seems incorrect
 
Sorry for the delayed response here, but i still find that

m_{H}(N) = 2 \sum_{i=0}^d \left({N-1}\atop i\right)

is not the case for a perceptron in d-dimensional space.

When N=6 and d=2, this equation says m_{H}(n) = 32, while i was able to get 38 dichotomies (by picking 6 equidistant points on the circumference of a circle).

If i am missing something, then can you please point me to the proof.

Thank you.

vsthakur 08-29-2012 03:10 AM

Problem 2.10
 
To prove : m_H(2N) \le m_H(N)^2

As this is a generic statement, it has to apply to every growth function. But all we know about the growth functions (in general) is their bound, in terms of N and d_{vc}.

Also, we know that if N \le d_{vc} then m_H(N) is an increasing function whose value is 2^N. But, if N > d_{vc}, then we can only say that m_H(N) is non-decreasing and is bounded by N^{d_{vc}} + 1.

I guess my question is that how can we prove the generic statement above. Kindly shed some light on the proof strategy.

Thank you,

Vishwajeet.

vsthakur 08-29-2012 07:46 AM

Possible correction to Problem 2.14 (b)
 
The problem says,

(b) Suppose that l satisfies 2^l  > K l^{d_{vc}+1}. Show that d_{vc}(H) \le l.

I suppose it should say,

(b) Suppose that l satisfies 2^l  > l^{K(d_{vc}+1)}. Show that d_{vc}(H) \le l.

If this is indeed the case, then can you please clarify part (c) as well.

Thank you,

Vishwajeet.

magdon 08-29-2012 09:30 AM

Re: Possible correction to Problem 2.14 (b)
 
The problem, though an over-estimate seems correct.

Hint: If you have \ell points, then {\cal H}_1 can implement at most \ell^{d_{VC}}+1\le\ell^{d_{VC}+1} dichotomies on those points. Now try to upper bound the number of dichotomies that all K hypothesis sets can implement on these \ell points and proceed from there.

Quote:

Originally Posted by vsthakur (Post 4579)
The problem says,

(b) Suppose that l satisfies 2^l  > K l^{d_{vc}+1}. Show that d_{vc}(H) \le l.

I suppose it should say,

(b) Suppose that l satisfies 2^l  > l^{K(d_{vc}+1)}. Show that d_{vc}(H) \le l.

If this is indeed the case, then can you please clarify part (c) as well.

Thank you,

Vishwajeet.



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