Exercises and Problems

yaser · #1 03-25-2012, 12:25 AM

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 · #2 07-02-2012, 02:36 PM

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 · #3 07-02-2012, 03:58 PM

Quote:

Originally Posted by tadworthington

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 · #4 08-24-2012, 09:20 AM

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 · #5 08-24-2012, 01:17 PM

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

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 · #6 08-25-2012, 09:04 PM

Thank you

vsthakur · #7 08-29-2012, 02:39 AM

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 · #8 08-29-2012, 03:10 AM

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

and $d_{vc}$ .

Also, we know that if $N \le d_{vc}$ then

is an increasing function whose value is

. But, if $N > d_{vc}$ , then we can only say that

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 · #9 08-29-2012, 07:46 AM

The problem says,

(b) Suppose that

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

I suppose it should say,

(b) Suppose that

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 · #10 08-29-2012, 09:30 AM

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

hypothesis sets can implement on these $\ell$ points and proceed from there.

Quote:

Originally Posted by vsthakur

The problem says,

(b) Suppose that

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

I suppose it should say,

(b) Suppose that

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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