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Introduction

In the last lecture , we studied bounds of the following form: for any δ > 0 , with probability at least 1 - δ ,

R ( f ) R ^ n ( f ) + log | F | + log 1 δ 2 n , f F

which led to upper bounds on the estimation error of the form

E [ R ( f ^ n ) ] - min f F R ( f ) log | F | + log ( n ) + 2 n .

The key assumptions made in deriving the error bounds were:

  • bounded loss function
  • finite collection of candidate functions

The bounds are valid for every P X Y and are called distribution-free.

Deriving bounds for countably infinite spaces

In this lecture we will generalize the previous results in a powerful way by developing bounds applicable to possibly infinitecollections of candidates. To start let us suppose that F is a countable, possibly infinite, collection of candidate functions.Assign a positive number c( f ) to each f F , such that

f F e - c ( f ) < .

The numbers c( f ) can be interpreted as

  • measures of complexity
  • -log of prior probabilities
  • codelengths

In particular, if P( f ) is the prior probability of f then

e - - log p ( f ) = p ( f )

so c ( f ) - log p ( f ) produces

f F e - c ( f ) = f F p ( f ) = 1 .

Now recall Hoeffding's inequality. For each f and every ϵ > 0

P R ( f ) - R ^ n ( f ) ϵ e - 2 n ϵ 2

or for every δ > 0

P R ( f ) - R ^ n ( f ) log 1 δ 2 n δ .

Suppose δ > 0 is specified. Using the values c( f ) for f F , define

δ ( f ) = e - c ( f ) δ .

Then we have

P R ( f ) - R ^ n ( f ) log 1 δ ( f ) 2 n δ ( f ) .

Furthermore we can apply the union bound as follows

P sup f F R ( f ) - R ^ n ( f ) - log ( 1 / δ ( f ) ) 2 n 0 P f F R ( f ) - R ^ n ( f ) log 1 δ ( f ) 2 n f F P R ( f ) - R ^ n ( f ) log 1 δ ( f ) 2 n f F δ ( f ) = f F e - c ( f ) δ = δ .

So for any δ > 0 with probability at least 1 - δ , we have that f F

R ( f ) R ^ n ( f ) + log 1 δ ( f ) 2 n = R ^ n ( f ) + c ( f ) + log 1 δ 2 n .

Special case

Suppose F is finite and c ( f ) = log | F | f F . Then

f F e - c ( f ) = f F e - log | F | = f F 1 | F | = 1

and

δ ( f ) = δ | F |

which implies that for any δ > 0 with probability at least 1 - δ , we have

R ( f ) R ^ n ( f ) + log | F | + log 1 δ ( f ) 2 n , f F .

Note that this is precisely the bound we derived in the last lecture .

Choosing c( f )

The generalized bounds allow us to handle countably infinite collections of candidate functions, but we require that

f F e - c ( f ) < .

Of course, if c ( f ) = - log p ( f ) where p ( f ) is a proper prior probability distribution then we have

f F e - c ( f ) = 1 .

However, it may be difficult to design a probability distribution over an infinite class of candidates. The coding perspectiveprovides a very practical means to this end.

Assume that we have assigned a uniquely decodable binary code to each f F , and let c( f ) denote the codelength for f . That is, the code for f is c( f ) bits long. A very useful class of uniquely decodable codes are called prefix codes.

Prefix Code
A code is called a prefix codeif no codeword is a prefix of any other codeword.

The kraft inequality

For any binary prefix code, the codeword lengths c 1 , c 2 , ... satisfy

i = 1 2 - c i 1 .

Conversely, given any c 1 , c 2 , ... satisfying the inequality above we can construct a prefix code with these codeword lengths.We will prove this result a bit later, but now let's see how this is useful in our learning problem.

Assume that we have assigned a binary prefix codeword to each f F , and let c( f ) denote the bit-length of the codeword for f . Set δ ( f ) = 2 - c ( f ) δ . Then

P f F R ( f ) - R ^ n ( f ) log 1 δ ( f ) 2 n f F P R ( f ) - R ^ n ( f ) log 1 δ ( f ) 2 n f F δ ( f ) = f F 2 - c ( f ) δ = δ .

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
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