<< Chapter < Page Chapter >> Page >

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

how to create a software using Android phone
Wiseman Reply
how
basra
what is the difference between C and C++.
Yan Reply
what is software
Sami Reply
software is a instructions like programs
Shambhu
what is the difference between C and C++.
Yan
yes, how?
Hayder
what is software engineering
Ahmad
software engineering is a the branch of computer science deals with the design,development, testing and maintenance of software applications.
Hayder
who is best bw software engineering and cyber security
Ahmad
Both software engineering and cybersecurity offer exciting career prospects, but your choice ultimately depends on your interests and skills. If you enjoy problem-solving, programming, and designing software syste
Hayder
what's software processes
Ntege Reply
I haven't started reading yet. by device (hardware) or for improving design Lol? Here. Requirement, Design, Implementation, Verification, Maintenance.
Vernon
I can give you a more valid answer by 5:00 By the way gm.
Vernon
it is all about designing,developing, testing, implementing and maintaining of software systems.
Ehenew
hello assalamualaikum
Sami
My name M Sami I m 2nd year student
Sami
what is the specific IDE for flutter programs?
Mwami Reply
jegudgdtgd my Name my Name is M and I have been talking about iey my papa john's university of washington post I tagged I will be in
Mwaqas Reply
yes
usman
how disign photo
atul Reply
hlo
Navya
hi
Michael
yes
Subhan
Show the necessary steps with description in resource monitoring process (CPU,memory,disk and network)
samuel Reply
What is software engineering
Tafadzwa Reply
Software engineering is a branch of computer science directed to writing programs to develop Softwares that can drive or enable the functionality of some hardwares like phone , automobile and others
kelvin
if any requirement engineer is gathering requirements from client and after getting he/she Analyze them this process is called
Alqa Reply
The following text is encoded in base 64. Ik5ldmVyIHRydXN0IGEgY29tcHV0ZXIgeW91IGNhbid0IHRocm93IG91dCBhIHdpbmRvdyIgLSBTdGV2ZSBXb3puaWFr Decode it, and paste the decoded text here
Julian Reply
what to do you mean
Vincent
hello
ALI
how are you ?
ALI
What is the command to list the contents of a directory in Unix and Unix-like operating systems
George Reply
how can i make my own software free of cost
Faizan Reply
like how
usman
hi
Hayder
The name of the author of our software engineering book is Ian Sommerville.
Doha Reply
what is software
Sampson Reply
the set of intruction given to the computer to perform a task
Noor
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistical learning theory' conversation and receive update notifications?

Ask