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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

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
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 .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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what is hormones?
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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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