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ε ( h ) = P ( x , y ) D ( h ( x ) y ) .

I.e. this is the probability that, if we now draw a new example ( x , y ) from the distribution D , h will misclassify it.

Note that we have assumed that the training data was drawn from the same distribution D with which we're going to evaluate our hypotheses (in the definition of generalization error). This issometimes also referred to as one of the PAC assumptions. PAC stands for “probably approximately correct,” which is a framework and set of assumptionsunder which numerous results on learning theory were proved. Of these, the assumption of training and testing on the same distribution, and the assumptionof the independently drawn training examples, were the most important.

Consider the setting of linear classification, and let h θ ( x ) = 1 { θ T x 0 } . What's a reasonable way of fitting the parameters θ ? One approach is to try to minimize the training error, and pick

θ ^ = arg min θ ε ^ ( h θ ) .

We call this process empirical risk minimization (ERM), and the resulting hypothesis output by the learning algorithm is h ^ = h θ ^ . We think of ERM as the most “basic” learning algorithm, and it will be thisalgorithm that we focus on in these notes. (Algorithms such as logistic regression can also be viewed as approximations to empirical risk minimization.)

In our study of learning theory, it will be useful to abstract away from the specific parameterization of hypotheses and from issues suchas whether we're using a linear classifier. We define the hypothesis class H used by a learning algorithm to be the set of all classifiers considered by it. For linearclassification, H = { h θ : h θ ( x ) = 1 { θ T x 0 } , θ R n + 1 } is thus the set of all classifiers over X (the domain of the inputs) where the decision boundary is linear.More broadly, if we were studying, say, neural networks, then we could let H be the set of all classifiers representable by some neural network architecture.

Empirical risk minimization can now be thought of as a minimization over the class of functions H , in which the learning algorithm picks the hypothesis:

h ^ = arg min h H ε ^ ( h )

The case of finite H

Let's start by considering a learning problem in which we have a finite hypothesis class H = { h 1 , ... , h k } consisting of k hypotheses. Thus, H is just a set of k functions mapping from X to { 0 , 1 } , and empirical risk minimization selects h ^ to be whichever of these k functions has the smallest training error.

We would like to give guarantees on the generalization error of h ^ . Our strategy for doing so will be in two parts: First, we will show that ε ^ ( h ) is a reliable estimate of ε ( h ) for all h . Second, we will show that this implies an upper-bound on the generalization error of h ^ .

Take any one, fixed, h i H . Consider a Bernoulli random variable Z whose distribution is defined as follows. We're going to sample ( x , y ) D . Then, we set Z = 1 { h i ( x ) y } . I.e., we're going to draw one example, and let Z indicate whether h i misclassifies it. Similarly, we also define Z j = 1 { h i ( x ( j ) ) y ( j ) } . Since our training set was drawn iid from D , Z and the Z j 's have the same distribution.

We see that the misclassification probability on a randomly drawn example—that is, ε ( h ) —is exactly the expected value of Z (and Z j ). Moreover, the training error can be written

Questions & Answers

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
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for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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?
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor 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, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
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