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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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