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ε ^ ( h i ) = 1 m j = 1 m Z j .

Thus, ε ^ ( h i ) is exactly the mean of the m random variables Z j that are drawn iid from a Bernoulli distribution with mean ε ( h i ) . Hence, we can apply the Hoeffding inequality, and obtain

P ( | ε ( h i ) - ε ^ ( h i ) | > γ ) 2 exp ( - 2 γ 2 m ) .

This shows that, for our particular h i , training error will be close to generalization error with high probability, assuming m is large. But we don't just want to guarantee that ε ( h i ) will be close to ε ^ ( h i ) (with high probability) for just only one particular h i . We want to prove that this will be true for simultaneously for all h H . To do so, let A i denote the event that | ε ( h i ) - ε ^ ( h i ) | > γ . We've already show that, for any particular A i , it holds true that P ( A i ) 2 exp ( - 2 γ 2 m ) . Thus, using the union bound, we have that

P ( h H . | ε ( h i ) - ε ^ ( h i ) | > γ ) = P ( A 1 A k ) i = 1 k P ( A i ) i = 1 k 2 exp ( - 2 γ 2 m ) = 2 k exp ( - 2 γ 2 m )

If we subtract both sides from 1, we find that

P ( ¬ h H . | ε ( h i ) - ε ^ ( h i ) | > γ ) = P ( h H . | ε ( h i ) - ε ^ ( h i ) | γ ) 1 - 2 k exp ( - 2 γ 2 m )

(The “ ¬ ” symbol means “not.”) So, with probability at least 1 - 2 k exp ( - 2 γ 2 m ) , we have that ε ( h ) will be within γ of ε ^ ( h ) for all h H . This is called a uniform convergence result, because this is a bound that holds simultaneously for all (as opposed to just one) h H .

In the discussion above, what we did was, for particular values of m and γ , give a bound on the probability that for some h H , | ε ( h ) - ε ^ ( h ) | > γ . There are three quantities of interest here: m , γ , and the probability of error; we can bound either one in terms of the other two.

For instance, we can ask the following question: Given γ and some δ > 0 , how large must m be before we can guarantee that with probability at least 1 - δ , training error will be within γ of generalization error? By setting δ = 2 k exp ( - 2 γ 2 m ) and solving for m , [you should convince yourself this is the right thing to do!], we find that if

m 1 2 γ 2 log 2 k δ ,

then with probability at least 1 - δ , we have that | ε ( h ) - ε ^ ( h ) | γ for all h H . (Equivalently, this shows that the probability that | ε ( h ) - ε ^ ( h ) | > γ for some h H is at most δ .) This bound tells us how many training examples we need in order makea guarantee. The training set size m that a certain method or algorithm requires in order to achieve a certain level of performance is also calledthe algorithm's sample complexity .

The key property of the bound above is that the number of training examples needed to make this guarantee is only logarithmic in k , the number of hypotheses in H . This will be important later.

Similarly, we can also hold m and δ fixed and solve for γ in the previous equation, and show [again, convince yourself that this is right!]that with probability 1 - δ , we have that for all h H ,

| ε ^ ( h ) - ε ( h ) | 1 2 m log 2 k δ .

Now, let's assume that uniform convergence holds, i.e., that | ε ( h ) - ε ^ ( h ) | γ for all h H . What can we prove about the generalization of our learning algorithm that picked h ^ = arg min h H ε ^ ( h ) ?

Define h * = arg min h H ε ( h ) to be the best possible hypothesis in H . Note that h * is the best that we could possibly do given that we are using H , so it makes sense to compare our performance to that of h * . We have:

ε ( h ^ ) ε ^ ( h ^ ) + γ ε ^ ( h * ) + γ ε ( h * ) + 2 γ

The first line used the fact that | ε ( h ^ ) - ε ^ ( h ^ ) | γ (by our uniform convergence assumption). The second used the fact that h ^ was chosen to minimize ε ^ ( h ) , and hence ε ^ ( h ^ ) ε ^ ( h ) for all h , and in particular ε ^ ( h ^ ) ε ^ ( h * ) . The third line used the uniform convergence assumption again, to show that ε ^ ( h * ) ε ( h * ) + γ . So, what we've shown is the following: If uniform convergence occurs,then the generalization error of h ^ is at most 2 γ worse than the best possible hypothesis in H !

Questions & Answers

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
characteristics of micro business
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
what is the k.e before it land
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
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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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