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Instructor (Andrew Ng): Oh, okay. Let me just – well, let me write that down on this board. So actually – actually let me think – [inaudible] fit this in here. So epsilon hat of H is the training error of the hypothesis H. In other words, given the hypothesis – a hypothesis is just a function, right mapped from X or Ys – so epsilon hat of H is given the hypothesis H, what’s the fraction of training examples it misclassifies? And generalization error of H, is given the hypothesis H if I sample another example from my distribution scripts D, what’s the probability that H will misclassify that example? Does that make sense?

Student: [Inaudible]?

Instructor (Andrew Ng): Oh, okay. And H hat is the hypothesis that’s chosen by empirical risk minimization. So when I talk about empirical risk minimization, is the algorithm that minimizes training error, and so epsilon hat of H is the training error of H, and so H hat is defined as the hypothesis that out of all hypotheses in my class script H, the one that minimizes training error epsilon hat of H. Okay?

All right. Yeah?

Student: [Inaudible] H is [inaudible]a member of typical H, [inaudible] family right?

Instructor (Andrew Ng): Yes it is.

Student: So what happens with the generalization error [inaudible]?

Instructor (Andrew Ng): I’ll talk about that later. So let me tie all these things together into a theorem. Let there be a hypothesis class given with a finite set of K hypotheses, and let any M delta be fixed. Then – so I fixed M and delta, so this will be the error bound form of the theorem, right? Then, with probability at least one minus delta. We have that. The generalization error of H hat is less than or equal to the minimum over all hypotheses in set H epsilon of H, plus two times, plus that. Okay?

So to prove this, well, this term of course is just epsilon of H star. And so to prove this we set gamma to equal to that – this is two times the square root term. To prove this theorem we set gamma to equal to that square root term. Say that again?

Student: [Inaudible].

Instructor (Andrew Ng): Wait. Say that again?

Student: [Inaudible].

Instructor (Andrew Ng): Oh, yes. Thank you. That didn’t make sense at all. Thanks. Great. So set gamma to that square root term, and so we know equation one, right, from the previous board holds with probability one minus delta. Right. Equation one was the uniform conversions result right, that – well, IE. This is equation one from the previous board, right, so set gamma equal to this we know that we’ll probably use one minus delta this uniform conversions holds, and whenever that holds, that implies – you know, I guess – if we call this equation “star” I guess. And whenever uniform conversions holds, we showed again, on the previous boards that this result holds, that generalization error of H hat is less than two – generalization error of H star plus two times gamma. Okay? And so that proves this theorem.

So this result sort of helps us to quantify a little bit that bias variance tradeoff that I talked about at the beginning of – actually near the very start of this lecture. And in particular let’s say I have some hypothesis class script H, that I’m using, maybe as a class of all linear functions and linear regression, and logistic regression with just the linear features. And let’s say I’m considering switching to some new class H prime by having more features. So lets say this is linear and this is quadratic, so the class of all linear functions and the subset of the class of all quadratic functions, and so H is the subset of H prime. And let’s say I’m considering – instead of using my linear hypothesis class – let’s say I’m considering switching to a quadratic hypothesis class, or switching to a larger hypothesis class. Then what are the tradeoffs involved? Well, I proved this only for finite hypothesis classes, but we’ll see that something very similar holds for infinite hypothesis classes too. But the tradeoff is what if I switch from H to H prime, or I switch from linear to quadratic functions. Then epsilon of H star will become better because the best hypothesis in my hypothesis class will become better.

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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?
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.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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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