LFD Book Forum  

Go Back   LFD Book Forum > Book Feedback - Learning From Data > Chapter 2 - Training versus Testing

Reply
 
Thread Tools Display Modes
  #1  
Old 02-25-2013, 01:27 PM
cls2k cls2k is offline
Junior Member
 
Join Date: May 2012
Posts: 5
Default Exercise problem 2.4

I'm stuck at the exercise problem 2.4 behind the book despite the hint. My approach is to characterize the B(N+1,K) >= recursion as an upper bound on the lower order (in N) terms and then follow the approach of proving the Sauer lemma. However I'm stuck on constructing the "specific set" of dichotomies. I fail to see how the special property to this set (limiting the number of -1 as hinted) can make this proof go easier.

I'm not very good at mathematical proofs so any additional hints will be greatly appreciated.
Reply With Quote
  #2  
Old 02-25-2013, 01:57 PM
magdon's Avatar
magdon magdon is offline
RPI
 
Join Date: Aug 2009
Location: Troy, NY, USA.
Posts: 595
Default Re: Exercise problem 2.4

Here is a hint. Lets consider showing B(5,2)>=1+5; It suffices to list 6 dichotomies on 5 points such that no subset of size 2 is shattered.

Consider the following 6 dichotomies on 5 points:

[1 1 1 1 1] (zero -1s)

[-1 1 1 1 1] (5 ways of having one -1)
[1 -1 1 1 1]
[1 1 -1 1 1]
[1 1 1 -1 1]
[1 1 1 1 -1]

Can you show that no subset of 2 points is shattered? Hint: are there any two points that are classified -1,-1?

So, in general, you can guarantee that no subset of size k can be shattered if no subset of size k is classified all -1. This means that at most k-1 points are classified -1 by any dichotomy.

Quote:
Originally Posted by cls2k View Post
I'm stuck at the exercise problem 2.4 behind the book despite the hint. My approach is to characterize the B(N+1,K) >= recursion as an upper bound on the lower order (in N) terms and then follow the approach of proving the Sauer lemma. However I'm stuck on constructing the "specific set" of dichotomies. I fail to see how the special property to this set (limiting the number of -1 as hinted) can make this proof go easier.

I'm not very good at mathematical proofs so any additional hints will be greatly appreciated.
__________________
Have faith in probability
Reply With Quote
  #3  
Old 02-25-2013, 09:40 PM
cls2k cls2k is offline
Junior Member
 
Join Date: May 2012
Posts: 5
Default Re: Exercise problem 2.4

Thank you! this is such an awesome and elegant insight and now the proof is easy.
Reply With Quote
  #4  
Old 02-27-2013, 07:53 AM
BojanVujatovic BojanVujatovic is offline
Member
 
Join Date: Jan 2013
Posts: 13
Default Re: Exercise problem 2.4

Professor, thank you for this amazingly helpful hint!
Reply With Quote
  #5  
Old 03-05-2018, 07:30 AM
k_sze k_sze is offline
Member
 
Join Date: Dec 2016
Posts: 12
Default Re: Exercise problem 2.4

I still don't get it after reading this hint.

Could somebody PM me to help me out so I don't ruin it for everybody else?
Reply With Quote
  #6  
Old 03-15-2018, 08:50 AM
k_sze k_sze is offline
Member
 
Join Date: Dec 2016
Posts: 12
Default Re: Exercise problem 2.4

Never mind. I got it.

Put another way: think of a way to systematically enumerate a set of dichotomies that won't shatter k out of N points. That's your minimum B(N, k).
Reply With Quote
Reply

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off

Forum Jump


All times are GMT -7. The time now is 10:17 PM.


Powered by vBulletin® Version 3.8.3
Copyright ©2000 - 2018, Jelsoft Enterprises Ltd.
The contents of this forum are to be used ONLY by readers of the Learning From Data book by Yaser S. Abu-Mostafa, Malik Magdon-Ismail, and Hsuan-Tien Lin, and participants in the Learning From Data MOOC by Yaser S. Abu-Mostafa. No part of these contents is to be communicated or made accessible to ANY other person or entity.