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Overview of the learning problem

The fundamental problem in learning from data is proper Model Selection. As we have seen in the previous lectures, a model that istoo complex could overfit the training data (causing an estimation error) and a model that is too simple could be a bad approximation ofthe function that we are trying to estimate (causing an approximation error). The estimation error arises because of the fact that we do notknow the true joint distribution of data in the input and output space, and therefore we minimize the empirical risk (which, for eachcandidate model, is a random number depending on the data) and estimate the average risk again from the limited number of trainingsamples we have. The approximation error measures how well the functions in the chosen model space can approximate the underlyingrelationship between the output space on the input space, and in general improves as the “size” of our model space increases.

Lecture outline

In the preceding lectures, we looked at some solutions to deal with the overfitting problem. The basic approach followed was the Methodof Sieves, in which the complexity of the model space was chosen as a function of the number of training samples. In particular, both thedenoising and classification problems we looked at consider estimators based on histogram partitions. The size of the partition was anincreasing function of the number of training samples. In this lecture, we will refine our learning methods further introduce modelselection procedures that automatically adapt to the distribution of the training data, rather than basing the model class solely on thenumber of samples. This sort of adaptivity will play a major role in the design of more effective classifiers and denoising methods. Thekey to designing data-adaptive model selection procedures is obtaining useful upper bounds on the estimation error. To this end, we willintroduce the idea of “Probably Approximately Correct” learning methods.

Recap: method of sieves

The method of Sieves underpinned our approaches in the denoising problem and in the histogram classification problem. Recall that thebasic idea is to define a sequence of model spaces F 1 , F 2 , ...of increasing complexity, and then given the training data { X i , Y i } i = 1 n select a model according to

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

The choice of the model space F n (and hence the model complexity and structure) is determined completely by the sample size n , and does not depend on the (empirical) distribution of training data.This is a major limitation of the sieve method. In a nutshell, the method of sieves tells us to average the data in a certain way (e.g., over a partition of X ) based on the sample size, independent on the sample values themselves.

In general, learning basically comprises of two things:

  1. Averaging data to reduce variability
  2. Deciding where (or how) to average

Sieves basically force us to deal with (2) a priori (before we analyze the training data). This will lead to suboptimalclassifiers and estimators, in general. Indeed deciding where/how to average is the really interesting and fundamental aspect of learning;once this is decided we have effectively solved the learing problem. There are at least two possibilities for breaking the rigidity of themethod of sieves, as we shall see in the following section.

Questions & Answers

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.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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 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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