0.3 Introduction to complexity regularization  (Page 2/3)

 Page 2 / 3

Revisit the polynomial regression example (Lecture 2, Ex. 4) , and incorporate a penalty term $C\left(f\right)$ 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 $\mathcal{F}$ 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

${F}_{1},{\mathcal{F}}_{2},\cdots ,{\mathcal{F}}_{n},\cdots$

where $|{\mathcal{F}}_{i}|$ grows as $i\to \infty$ . 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 ${\left\{{X}_{i},{Y}_{i}\right\}}_{i=1}^{n}$ i.i.d. distributed according to ${P}_{XY}$ , select $f\in {\mathcal{F}}_{n}$ to minimize the empirical risk

${\stackrel{^}{f}}_{n}=arg\underset{f\in {\mathcal{F}}_{n}}{min}{\stackrel{^}{R}}_{n}\left(f\right).$

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 $n\to \infty$ . 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.

Bayesian methods

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

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

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 $c\left(f\right)$ 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

$\begin{array}{c}\hfill \underset{f\in \mathcal{F}}{min}\left\{{\stackrel{^}{R}}_{n},\left(f\right),+,C,\left(f\right)\right\}.\end{array}$

In the Bayesian setting, $p\left(f\right)\propto {e}^{-C\left(f\right)}$ can be interpreted as a prior probability density on $\mathcal{F}$ , 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 $\mathcal{F}$ relative to a random sample of $n$ training data. For example, take $\mathcal{F}$ 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.

what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!