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In this set of notes, we begin our foray into learning theory. Apart from being interesting and enlightening in its own right, this discussion will also help ushone our intuitions and derive rules of thumb about how to best apply learning algorithms in different settings. We will also seek to answer a few questions: First, can wemake formal the bias/variance tradeoff that was just discussed? The will also eventually lead us to talk about model selection methods, which can,for instance, automatically decide what order polynomial to fit to a training set. Second, in machine learning it's really generalization errorthat we care about, but most learning algorithms fit their models to the training set. Why should doing well on the training set tell us anythingabout generalization error? Specifically, can we relate error on the training set to generalization error? Third and finally, are there conditions underwhich we can actually prove that learning algorithms will work well?

We start with two simple but very useful lemmas.

Lemma. (The union bound). Let A 1 , A 2 , ... , A k be k different events (that may not be independent). Then

P ( A 1 A k ) P ( A 1 ) + ... + P ( A k ) .

In probability theory, the union bound is usually stated as an axiom (and thus we won't try to prove it), but it also makes intuitive sense: The probability ofany one of k events happening is at most the sums of the probabilities of the k different events.

Lemma. (Hoeffding inequality) Let Z 1 , ... , Z m be m independent and identically distributed (iid) random variables drawn from a Bernoulli( Φ ) distribution. I.e., P ( Z i = 1 ) = Φ , and P ( Z i = 0 ) = 1 - Φ . Let Φ = ( 1 / m ) i = 1 m Z i be the mean of these random variables, and let any γ > 0 be fixed. Then

P ( | Φ - Φ ^ | > γ ) 2 exp ( - 2 γ 2 m )

This lemma (which in learning theory is also called the Chernoff bound ) says that if we take Φ ^ —the average of m Bernoulli( Φ ) random variables—to be our estimate of Φ , then the probability of our being far from the true value is small, so long as m is large. Another way of saying this is that if you have a biased coin whosechance of landing on heads is Φ , then if you toss it m times and calculate the fraction of times that it came up heads, that will be agood estimate of Φ with high probability (if m is large).

Using just these two lemmas, we will be able to prove some of the deepest and most important results in learning theory.

To simplify our exposition, let's restrict our attention to binary classification in which the labels are y { 0 , 1 } . Everything we'll say here generalizes to other, including regression and multi-classclassification, problems.

We assume we are given a training set S = { ( x ( i ) , y ( i ) ) ; i = 1 , ... , m } of size m , where the training examples ( x ( i ) , y ( i ) ) are drawn iid from some probability distribution D . For a hypothesis h , we define the training error (also called the empirical risk or empirical error in learning theory) to be

ε ^ ( h ) = 1 m i = 1 m 1 { h ( x ( i ) ) y ( i ) } .

This is just the fraction of training examples that h misclassifies. When we want to make explicit the dependence of ε ^ ( h ) on the training set S , we may also write this a ε ^ S ( h ) . We also define the generalization error to be

Questions & Answers

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 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
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
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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