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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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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 .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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
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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