# 2.3 Machine learning lecture 4 course notes  (Page 2/5)

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## Preliminaries

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?

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

$P\left({A}_{1}\cup \cdots \cup {A}_{k}\right)\le P\left({A}_{1}\right)+...+P\left({A}_{k}\right).$

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( $\Phi$ ) distribution. I.e., $P\left({Z}_{i}=1\right)=\Phi$ , and $P\left({Z}_{i}=0\right)=1-\Phi$ . Let $\stackrel{‸}{\Phi }=\left(1/m\right){\sum }_{i=1}^{m}{Z}_{i}$ be the mean of these random variables, and let any $\gamma >0$ be fixed. Then

$P\left(|\Phi -\stackrel{^}{\Phi }|>\gamma \right)\le 2exp\left(-2{\gamma }^{2}m\right)$

This lemma (which in learning theory is also called the Chernoff bound ) says that if we take $\stackrel{^}{\Phi }$ —the average of $m$ Bernoulli( $\Phi$ ) random variables—to be our estimate of $\Phi$ , 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 $\Phi$ , then if you toss it $m$ times and calculate the fraction of times that it came up heads, that will be agood estimate of $\Phi$ 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 $y\in \left\{0,1\right\}$ . Everything we'll say here generalizes to other, including regression and multi-classclassification, problems.

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

$\stackrel{^}{\epsilon }\left(h\right)=\frac{1}{m}\sum _{i=1}^{m}1\left\{h\left({x}^{\left(i\right)}\right)\ne {y}^{\left(i\right)}\right\}.$

This is just the fraction of training examples that $h$ misclassifies. When we want to make explicit the dependence of $\stackrel{^}{\epsilon }\left(h\right)$ on the training set $S$ , we may also write this a ${\stackrel{^}{\epsilon }}_{S}\left(h\right)$ . We also define the generalization error to be

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yeah
Joseph
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no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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Rafiq
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Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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