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Minimum complexity penalized function

Recall the basic results of the last lectures: let X and Y denote the input and output spaces respectively. Let X X and Y X be random variables with unknown joint probability distribution P X Y . We would like to use X to “predict” Y . Consider a loss function 0 ( y 1 , y 2 ) 1 , y 1 , y 2 Y . This function is used to measure the accuracy of our prediction. Let F be a collection of candidate functions (models), f : X Y . The expected risk we incur is given by R ( f ) E X Y [ ( f ( X ) , Y ) ] . We have access only to a number of i.i.d. samples, { X i , Y i } i = 1 n . These allow us to compute the empirical risk R ^ n ( f ) 1 n i = 1 n ( f ( X i ) , Y i ) .

Assume in the following that F is countable. Assign a positive number c ( f ) to each f F such that f F 2 - c ( f ) 1 . If we use a prefix code to describe each element of F and define c ( f ) to be the codeword length (in bits) for each f F , the last inequality is automatically satisfied.

We define the minimum complexity penalized estimator as

f ^ n arg min f F R ^ n ( f ) + c ( f ) log 2 + 1 2 log n 2 n .

As we showed previously we have the bound

E [ R ( f ^ n ) ] min f F R ( f ) + c ( f ) log 2 + 1 2 log n 2 n + 1 n .

The performance (risk) of f ^ n is on average better than

R ( f n * ) + c ( f n * ) log 2 + 1 2 log n 2 n + 1 n ,

where

f n * = arg min f F R ( f ) + c ( f ) log 2 + 1 2 log n 2 n .

If it happens that the optimal function, that is

f * = arg min f measurable R ( f ) ,

is close to an f F with a small c ( f ) , then f ^ n will perform almost as well as the optimal function.

Suppose f * F , then

E [ R ( f ^ n ) ] R ( f * ) + c ( f * ) log 2 + 1 2 log n 2 n + 1 n .

Furthermore if c ( f * ) = O ( log n ) then

E [ R ( f ^ n ) ] R ( f * ) + O log n n ,

that is, only within a small O log n n offset of the optimal risk.

In general, we can also bound the excess risk E [ R ( f ^ n ) ] - R * , where R * is the Bayes risk,

R * = inf f measurable R ( f ) .

By subtracting R * (a constant) from both sides of the inequality

E [ R ( f ^ n ) ] min f F R ( f ) + c ( f ) log 2 + 1 2 log n 2 n + 1 n

we obtain

E [ R ( f ^ n ) ] - R * min f F R ( f ) - R * + c ( f ) log 2 + 1 2 log n 2 n + 1 n .

Note that two terms in this upper bound: R ( f ) - R * is a bound on the approximation error of a model f , and remainder is a bound on the estimation error associated with f . Thus, we see that complexity regularization automatically optimizes a balance between approximation and estimationerrors. In other words, complexity regularization is adaptive to the unknown tradeoff between approximation and estimation.

Classification

Consider the particularization of the above to a classification scenario. Let X = [ 0 , 1 ] d , Y = { 0 , 1 } and ( y ^ , y ) 1 { y ^ y } . Then R ( f ) = E X Y [ 1 { f ( X ) Y } ] = P ( f ( X ) Y ) . The Bayes risk is given by

R * = inf f measurable R ( f ) .

As it was observed before, the Bayes classifier ( i.e., a classifier that achieves the Bayes risk) is given by

f * ( x ) = 1 , P ( Y = 1 | X = x ) 1 2 0 , P ( Y = 1 | X = x ) < 1 2 .

This classifier can be expressed in a different way. Consider the set G * = { x : P ( Y = 1 | X = x ) 1 / 2 } . The Bayes classifier can written as f * ( x ) = 1 { x G * } . Therefore the classifier is characterized entirely by the set G * , if X G * then the “best” guess is that Y is one, and vice-versa. The boundary of this set corresponds to the points where the decision is harder.The boundary of G * is called the Bayes Decision Boundary . In [link] (a) this concept is illustrated. If η ( x ) = P ( Y = 1 | X = x ) is a continuous function then the Bayes decision boundary is simply given by { x : P ( Y = 1 | X = x ) = 1 / 2 } . Clearly the structure of the decision boundary provides importantinformation on the difficulty of the problem.

Questions & Answers

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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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
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
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
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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