# 0.6 Probably approximately correct (pac) learning

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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 ${\mathcal{F}}_{1}$ , ${\mathcal{F}}_{2}$ , ...of increasing complexity, and then given the training data ${\left\{{X}_{i},{Y}_{i}\right\}}_{i=1}^{n}$ select a model according to

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

The choice of the model space ${\mathcal{F}}_{n}$ (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 $\mathcal{X}$ ) 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.

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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
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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
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it is a goid question and i want to know the answer as well
Maciej
Abigail
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Anassong
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
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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