# 2.2 Machine learning lecture 3 course notes  (Page 2/12)

 Page 2 / 12

## Notation

To make our discussion of SVMs easier, we'll first need to introduce a new notation for talking about classification.We will be considering a linear classifier for a binary classification problem with labels $y$ and features $x$ . From now, we'll use $y\in \left\{-1,1\right\}$ (instead of $\left\{0,1\right\}$ ) to denote the class labels. Also, rather than parameterizing our linearclassifier with the vector $\theta$ , we will use parameters $w,b$ , and write our classifier as

${h}_{w,b}\left(x\right)=g\left({w}^{T}x+b\right).$

Here, $g\left(z\right)=1$ if $z\ge 0$ , and $g\left(z\right)=-1$ otherwise. This “ $w,b$ ” notation allows us to explicitly treat the intercept term $b$ separately from the other parameters. (We also drop the convention we had previously of letting ${x}_{0}=1$ be an extra coordinate in the input feature vector.) Thus, $b$ takes the role of what was previously ${\theta }_{0}$ , and $w$ takes the role of ${\left[{\theta }_{1}...{\theta }_{n}\right]}^{T}$ .

Note also that, from our definition of $g$ above, our classifier will directly predict either 1 or $-1$ (cf. the perceptron algorithm), without first going through the intermediate step of estimating the probability of $y$ being 1 (which was what logistic regression did).

## Functional and geometric margins

Let's formalize the notions of the functional and geometric margins. Given a training example $\left({x}^{\left(i\right)},{y}^{\left(i\right)}\right)$ , we define the functional margin of $\left(w,b\right)$ with respect to the training example

$\stackrel{^}{\gamma }{}^{\left(i\right)}={y}^{\left(i\right)}\left({w}^{T}x+b\right).$

Note that if ${y}^{\left(i\right)}=1$ , then for the functional margin to be large (i.e., for our prediction to be confident and correct), we need ${w}^{T}x+b$ to be a large positive number. Conversely, if ${y}^{\left(i\right)}=-1$ , then for the functional margin to be large, we need ${w}^{T}x+b$ to be a large negative number. Moreover, if ${y}^{\left(i\right)}\left({w}^{T}x+b\right)>0$ , then our prediction on this example is correct. (Check this yourself.)Hence, a large functional margin represents a confident and a correct prediction.

For a linear classifier with the choice of $g$ given above (taking values in $\left\{-1,1\right\}$ ), there's one property of the functional margin that makes it not a very good measure of confidence,however. Given our choice of $g$ , we note that if we replace $w$ with $2w$ and $b$ with $2b$ , then since $g\left({w}^{T}x+b\right)=g\left(2{w}^{T}x+2b\right)$ , this would not change ${h}_{w,b}\left(x\right)$ at all. I.e., $g$ , and hence also ${h}_{w,b}\left(x\right)$ , depends only on the sign, but not on the magnitude, of ${w}^{T}x+b$ . However, replacing $\left(w,b\right)$ with $\left(2w,2b\right)$ also results in multiplying our functional margin by a factor of 2. Thus, it seems that by exploiting our freedom to scale $w$ and $b$ , we can make the functional margin arbitrarily large without really changing anything meaningful. Intuitively, it might therefore make senseto impose some sort of normalization condition such as that ${||w||}_{2}=1$ ; i.e., we might replace $\left(w,b\right)$ with $\left(w/||w|{|}_{2},b/||w|{|}_{2}\right)$ , and instead consider the functional margin of $\left(w/||w|{|}_{2},b/||w|{|}_{2}\right)$ . We'll come back to this later.

Given a training set $S=\left\{\left({x}^{\left(i\right)},{y}^{\left(i\right)}\right);i=1,...,m\right\}$ , we also define the function margin of $\left(w,b\right)$ with respect to $S$ to be the smallest of the functional margins of the individual training examples. Denotedby $\stackrel{^}{\gamma }$ , this can therefore be written:

$\stackrel{^}{\gamma }=\underset{i=1,...,m}{min}\stackrel{^}{\gamma }{}^{\left(i\right)}.$

Next, let's talk about geometric margins . Consider the picture below:

The decision boundary corresponding to $\left(w,b\right)$ is shown, along with the vector $w$ . Note that $w$ is orthogonal (at ${90}^{\circ }$ ) to the separating hyperplane. (You should convince yourself that this must be the case.) Consider the point at A, which represents the input ${x}^{\left(i\right)}$ of some training example with label ${y}^{\left(i\right)}=1$ . Its distance to the decision boundary, ${\gamma }^{\left(i\right)}$ , is given by the line segment AB.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!