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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
what is the stm
Brian Reply
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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?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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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 .
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.
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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