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Recap: classifier design

Given a set of training data { X i , Y i } i = 1 n and a finite collection of candidate functions F , select f ^ n F that (hopefully) is a good predictor for future cases. That is

f n ^ = arg min f F R ^ n ( f )

where R ^ n ( f ) is the empirical risk. For any particular f F , the corresponding empirical risk is defined as

R ^ n ( f ) = 1 n i = 1 n 1 { f ( X i ) Y i } .

Hoeffding's inequality

Hoeffding's inequality (Chernoff's bound in this case) allows us to gauge how close R ^ n ( f ) is to the true risk of f , R ( f ) , in probability

P ( | R ^ n ( f ) - R ( f ) | ϵ ) 2 e - 2 n ϵ 2 .

Since our selection process involves deciding among all f F , we would like to gauge how close the empirical risks are to theirexpected values. We can do this by studying the probability that one or more of the empirical risks deviates significantly from itsexpected value. This is captured by the probability

P max f F | R ^ n ( f ) - R ( f ) | ϵ .

Note that the event

max f F | R ^ n ( f ) - R ( f ) | ϵ

is equivalent to union of the events

f F | R ^ n ( f ) - R ( f ) | ϵ .

Therefore, we can use Bonferonni's bound (aka the “union of events” or “union” bound) to obtain

P max f F | R ^ n ( f ) - R ( f ) | ϵ = P f F | R ^ n ( f ) - R ( f ) | ϵ f F P ( | R ^ n ( f ) - R ( f ) | ϵ ) f F 2 e - 2 n ϵ 2 = 2 | F | e - 2 n ϵ 2

where | F | is the number of classifiers in F . In the proof of Hoeffding's inequality we also obtained a one-sided inequality thatimplied

P ( R ( f ) - R ^ n ( f ) ϵ ) e - 2 n ϵ 2

and hence

P max f F R ( f ) - R ^ n ( f ) ϵ | F | e - 2 n ϵ 2 .

We can restate the inequality above as follows, For all f F and for all δ > 0 with probability at least 1 - δ

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

This follows by setting δ = | F | e - 2 n ϵ 2 and solving for ϵ . Thus with a high probability ( 1 - δ ) , the true risk for all f F is bounded by the empirical risk of f plus a constant that depends on δ > 0 , the number of training samples n, and the size F . Most importantly the bound does not depend on the unknown distribution P X Y . Therefore, we can call this a distribution-free bound.

Error bounds

We can use the distribution-free bound above to obtain a bound on the expected performance of the minimum empirical riskclassifier

f ^ n = arg min f F R ^ n ( f ) .

We are interested in bounding

E [ R ( f ^ n ) ] - min f F R ( f )

the expected risk of f ^ n minus the minimum risk for all f F . Note that this difference is always non-negative since f ^ n is at best as good as

f * = arg min f F R ( f ) .

Recall that f F and δ > 0 , with probability at least 1 - δ

R ( f ) R ^ n ( f ) + C ( F , n , δ )

where

C ( F , n , δ ) = log | F | + log ( 1 / δ ) 2 n .

In particular, since this holds for all f F including f ^ n ,

R ( f ^ n ) R ^ n ( f ^ n ) + C ( F , n , δ )

and for any other f F

R ( f ^ n ) R ^ n ( f ) + C ( F , n , δ )

since R ^ n ( f ^ n ) R ^ n ( f ) f F . In particular,

R ( f ^ n ) R ^ n ( f * ) + C ( F , n , δ )

where f * = arg min f F R ( f ) .

Let Ω denote the set of events on which the above inequality holds. Then by definition

P ( Ω ) 1 - δ .

We can now bound E [ R ( f ^ n ) ] - R ( f * ) as follows

E [ R ( f ^ n ) ] - R ( f * ) = E [ R ( f ^ n ) - R ^ n ( f * ) + R ^ n ( f * ) - R ( f * ) ] = E [ R ( f ^ n ) - R ^ n ( f * ) ]

since E [ R ^ n ( f * ) ] = R ( f * ) . The quantity above is bounded as follows.

E [ R ( f ^ n ) - R ^ n ( f * ) ] = E [ R ( f ^ n ) - R ^ n ( f * ) | Ω ] P ( Ω ) + E [ R ( f ^ n ) - R ^ n ( f * ) | Ω ] P ( Ω ¯ ) E [ R ( f ^ n ) - R ^ n ( f * ) | Ω ] + δ

since P ( Ω ) 1 , 1 - P ( Ω ) δ and R ( f ^ n ) - R ^ n ( f * ) 1

E [ R ( f ^ n ) - R ^ n ( f * ) | Ω ] E [ R ( f ^ n ) - R ^ n ( f ^ n ) | Ω ] C ( F , n , δ ) .

Thus

E [ R ( f ^ n ) - R ^ n ( f * ) ] C ( F , n , δ ) + δ .

So we have

E [ R ( f ^ n ) ] - min f F R ( f ) log | F | + log ( 1 / δ ) 2 n + δ , δ > 0 .

In particular, for δ = 1 / n , we have

E [ R ( f ^ n ) ] - min f F R ( f ) log | F | + log n 2 n + 1 n log | F | + log n + 2 n , since x + y 2 x + y , x , y > 0 .

Application: histogram classifier

Let F be the collection of all classifiers with M equal volume cells. Then | F | = 2 M , and the histogram classification rule

f ^ n = arg min f F 1 n i = 1 n 1 { f ( X i ) Y i }

satisfies

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

which suggests the choice M = log 2 n (balancing M log 2 with log n ), resulting in

E [ R ( f ^ n ) ] - min f F R ( f ) = O log n n .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
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
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
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
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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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