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Revisit the polynomial regression example (Lecture 2, Ex. 4) , and incorporate a penalty term C ( f ) which is proportional to the degree of f , or the derivative of f . In essence, this approach penalizes for functions which are too “wiggly”, with theintuition being that the true function is probably smooth so a function which is very wiggly will overfit the data.

How do we decide how to restrict or penalize the empirical risk minimization process? Approaches which have appeared in theliterature include the following.

Method of sieves

Perhaps the simplest approach is to try to limit the size of F in a way that depends on the number of training data n . The more data we have, the more complex the space of models we can entertain.Let the class of candidate functions grow with n . That is, take

F 1 , F 2 , , F n ,

where | F i | grows as i . In other words, consider a sequence of spaces with increasing complexity ordegrees of freedom depending on the number of training data samples, n .

Given samples { X i , Y i } i = 1 n i.i.d. distributed according to P X Y , select f F n to minimize the empirical risk

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

In the next lecture we will consider an example using the method of sieves. The basic idea is to design the sequence of model spaces in such a waythat the excess risk decays to zero as n . This sort of idea has been around for decades, but Grenander's method ofsieves is often cited as a nice formalization of the idea: Abstract Inference , Wiley, New York.

Complexity penalization methods

Bayesian methods

In certain cases, the empirical risk happens to be a (log) likelihood function, and one can then interpret the cost C ( f ) as reflecting prior knowledge about which models are more or less likely. In thiscase, e - C ( f ) is like a prior probability distribution on the space F . The cost C ( f ) is large if f is highly improbable, and C ( f ) is small if f is highly probable.

Alternatively, if we restrict F to be small, and denote the space of all measurable functions as F = F F c , then it is essentially as if we have placed a uniform prior over all functions in F , and zero prior probability on the functions in F c .

Description length methods

Description length methods represent each f with a string of bits. More complicated functions require more bits to represent.Accordingly, we can then set the cost c ( f ) proportional to the number of bits needed to describe f (the description length ). This results in what is known as the minimum description length (MDL)approach where the minimum description length is given by

min f F R ^ n ( f ) + C ( f ) .

In the Bayesian setting, p ( f ) e - C ( f ) can be interpreted as a prior probability density on F , with more complex models being less probable and simpler models being more probable. In that sense,both the Bayesian and MDL approaches have a similar spirit.

Vapnik-cervonenkis dimension

The Vapnik-Cervonenkis (VC) dimension measures the complexity of a class F relative to a random sample of n training data. For example, take F to be all linear classifiers in 2-dimensional feature space. Clearly, the space of linear classifiers isinfinite (there are an infinite number of lines which can be drawn in the plane). However, many of these linear classifiers would assignthe same labels to the training data.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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