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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

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.
what is biological synthesis of nanoparticles
Sanket Reply
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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