LFD Book Forum  

Go Back   LFD Book Forum > General > General Discussion of Machine Learning

 
 
Thread Tools Display Modes
Prev Previous Post   Next Post Next
  #1  
Old 02-16-2016, 07:29 PM
wordchao wordchao is offline
Junior Member
 
Join Date: Feb 2016
Posts: 2
Default example non-overlap hypothesis set H to make union bound equals vc bound

consider the union bound, when H consists of only h1 and h2,
Pr[|Ein(g)-Eout(g)|>e]
<=Pr[|Ein(h1)-Eout(h1)|>e or |Ein(h2)-Eout(h2)|>e] (eq.1)
<=Pr[|Ein(h1)-Eout(h1)|>e] + Pr[|Ein(h2)-Eout(h2)|>e] (eq.2)
=2exp(-2Ne^2) (eq.3)

i wonder which kind of h1 and h2 can more "accurately" satisfy the union bound, maybe this means union bound equals vc bound.
in other words, can i find h1 and h2 to let (eq.1)=(eq.2)?

Pr[|Ein(h1)-Eout(h1)|>e or |Ein(h2)-Eout(h2)|>e]
=Pr[|Ein(h1)-Eout(h1)|>e and |Ein(h2)-Eout(h2)|>e] (eq.a)
+Pr[|Ein(h1)-Eout(h1)|>e and |Ein(h2)-Eout(h2)|<=e] (eq.b)
+Pr[|Ein(h1)-Eout(h1)|<=e and |Ein(h2)-Eout(h2)|>e] (eq.c)

if (eq.2)=(eq.1), then,
sup(eq.a)=0
sup(eq.b)=sup(eq.c)=1/2(eq.3)=exp(-2Ne^2)

but i cannot find the example h1&h2, which means h1 and h2 has no overlap when considering the union bound.
can u help? thanks.
Reply With Quote
 

Tags
hypothesis, on-overlap, union bound

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 07:26 AM.


Powered by vBulletin® Version 3.8.3
Copyright ©2000 - 2020, 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.