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-   -   Problem with understanding the proof of Sauer Lemma (http://book.caltech.edu/bookforum/showthread.php?t=4604)

 yongxien 06-11-2015 12:31 AM

Problem with understanding the proof of Sauer Lemma

I will replicate the proof here which is from the book "Learning from Data"

Sauer Lemma:
$B(N,K) \leq \sum_{i=0}^{k-1}{n\choose i}$

Proof:
The statement is true whenever k = 1 or N = 1 by inspection. The proof is by induction on N. Assume the statement is true for all $N \leq N_o$ and for all k. We need to prove that the statement for $N = N_0 + 1$ and fpr all k. Since the statement is already true when k = 1(for all values of N) by the initial condition, we only need to worry about $k \geq 2$. By (proven in the book), $B(N_0 + 1, k) \leq B(N_0, k) + B(N_0, k-1)$ and applying induction hypothesis on each therm on the RHS, we get the result.

**My Concern** From what I see this proof only shows that if $B(N, K)$ implies $B(N+1, K)$. I can't see how it shows $B(N, K)$ implies $B(N, K+1)$. This problem arises because the $k$ in $B(N_0 + 1, K)$ and $B(N_0, K)$ are the same, so i think i need to prove the other induction too. Why the author is able to prove it this way?

 yongxien 06-11-2015 12:34 AM

Re: Problem with understanding the proof of Sauer Lemma

OK i think i will just post it below. I can't find an edit button. I mean for 2 variable induction, shouldn't we prove B(N,k) implies B(N+1,k) and B(N, K+1)?

 htlin 06-14-2015 08:26 PM

Re: Problem with understanding the proof of Sauer Lemma

You can imaging that the induction hypothesis to be satisfying the inequality for "all k", and then, satisfies the inequality for "all k" too.

Hope this helps.

 yongxien 07-23-2015 12:03 AM

Re: Problem with understanding the proof of Sauer Lemma

That is my concern. Why "all k" when we have not proved it.

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