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

 Page 4 / 5
$\stackrel{^}{\epsilon }\left({h}_{i}\right)=\frac{1}{m}\sum _{j=1}^{m}{Z}_{j}.$

Thus, $\stackrel{^}{\epsilon }\left({h}_{i}\right)$ is exactly the mean of the $m$ random variables ${Z}_{j}$ that are drawn iid from a Bernoulli distribution with mean $\epsilon \left({h}_{i}\right)$ . Hence, we can apply the Hoeffding inequality, and obtain

$P\left(|\epsilon \left({h}_{i}\right)-\stackrel{^}{\epsilon }\left({h}_{i}\right)|>\gamma \right)\le 2exp\left(-2{\gamma }^{2}m\right).$

This shows that, for our particular ${h}_{i}$ , training error will be close to generalization error with high probability, assuming $m$ is large. But we don't just want to guarantee that $\epsilon \left({h}_{i}\right)$ will be close to $\stackrel{^}{\epsilon }\left({h}_{i}\right)$ (with high probability) for just only one particular ${h}_{i}$ . We want to prove that this will be true for simultaneously for all $h\in \mathcal{H}$ . To do so, let ${A}_{i}$ denote the event that $|\epsilon \left({h}_{i}\right)-\stackrel{^}{\epsilon }\left({h}_{i}\right)|>\gamma$ . We've already show that, for any particular ${A}_{i}$ , it holds true that $P\left({A}_{i}\right)\le 2exp\left(-2{\gamma }^{2}m\right)$ . Thus, using the union bound, we have that

$\begin{array}{ccc}\hfill P\left(\exists \phantom{\rule{0.166667em}{0ex}}h\in \mathcal{H}.|\epsilon \left({h}_{i}\right)-\stackrel{^}{\epsilon }\left({h}_{i}\right)|>\gamma \right)& =& P\left({A}_{1}\cup \cdots \cup {A}_{k}\right)\hfill \\ & \le & \sum _{i=1}^{k}P\left({A}_{i}\right)\hfill \\ & \le & \sum _{i=1}^{k}2exp\left(-2{\gamma }^{2}m\right)\hfill \\ & =& 2kexp\left(-2{\gamma }^{2}m\right)\hfill \end{array}$

If we subtract both sides from 1, we find that

$\begin{array}{ccc}\hfill P\left(¬\exists \phantom{\rule{0.166667em}{0ex}}h\in \mathcal{H}.|\epsilon \left({h}_{i}\right)-\stackrel{^}{\epsilon }\left({h}_{i}\right)|>\gamma \right)& =& P\left(\forall h\in \mathcal{H}.|\epsilon \left({h}_{i}\right)-\stackrel{^}{\epsilon }\left({h}_{i}\right)|\le \gamma \right)\hfill \\ & \ge & 1-2kexp\left(-2{\gamma }^{2}m\right)\hfill \end{array}$

(The “ $¬$ ” symbol means “not.”) So, with probability at least $1-2kexp\left(-2{\gamma }^{2}m\right)$ , we have that $\epsilon \left(h\right)$ will be within $\gamma$ of $\stackrel{^}{\epsilon }\left(h\right)$ for all $h\in \mathcal{H}$ . This is called a uniform convergence result, because this is a bound that holds simultaneously for all (as opposed to just one) $h\in \mathcal{H}$ .

In the discussion above, what we did was, for particular values of $m$ and $\gamma$ , give a bound on the probability that for some $h\in \mathcal{H}$ , $|\epsilon \left(h\right)-\stackrel{^}{\epsilon }\left(h\right)|>\gamma$ . There are three quantities of interest here: $m$ , $\gamma$ , and the probability of error; we can bound either one in terms of the other two.

For instance, we can ask the following question: Given $\gamma$ and some $\delta >0$ , how large must $m$ be before we can guarantee that with probability at least $1-\delta$ , training error will be within $\gamma$ of generalization error? By setting $\delta =2kexp\left(-2{\gamma }^{2}m\right)$ and solving for $m$ , [you should convince yourself this is the right thing to do!], we find that if

$m\ge \frac{1}{2{\gamma }^{2}}log\frac{2k}{\delta },$

then with probability at least $1-\delta$ , we have that $|\epsilon \left(h\right)-\stackrel{^}{\epsilon }\left(h\right)|\le \gamma$ for all $h\in \mathcal{H}$ . (Equivalently, this shows that the probability that $|\epsilon \left(h\right)-\stackrel{^}{\epsilon }\left(h\right)|>\gamma$ for some $h\in \mathcal{H}$ is at most $\delta$ .) This bound tells us how many training examples we need in order makea guarantee. The training set size $m$ that a certain method or algorithm requires in order to achieve a certain level of performance is also calledthe algorithm's sample complexity .

The key property of the bound above is that the number of training examples needed to make this guarantee is only logarithmic in $k$ , the number of hypotheses in $\mathcal{H}$ . This will be important later.

Similarly, we can also hold $m$ and $\delta$ fixed and solve for $\gamma$ in the previous equation, and show [again, convince yourself that this is right!]that with probability $1-\delta$ , we have that for all $h\in \mathcal{H}$ ,

$|\stackrel{^}{\epsilon }\left(h\right)-\epsilon \left(h\right)|\le \sqrt{\frac{1}{2m}log\frac{2k}{\delta }}.$

Now, let's assume that uniform convergence holds, i.e., that $|\epsilon \left(h\right)-\stackrel{^}{\epsilon }\left(h\right)|\le \gamma$ for all $h\in \mathcal{H}$ . What can we prove about the generalization of our learning algorithm that picked $\stackrel{^}{h}=arg{min}_{h\in \mathcal{H}}\stackrel{^}{\epsilon }\left(h\right)$ ?

Define ${h}^{*}=arg{min}_{h\in \mathcal{H}}\epsilon \left(h\right)$ to be the best possible hypothesis in $\mathcal{H}$ . Note that ${h}^{*}$ is the best that we could possibly do given that we are using $\mathcal{H}$ , so it makes sense to compare our performance to that of ${h}^{*}$ . We have:

$\begin{array}{ccc}\hfill \epsilon \left(\stackrel{^}{h}\right)& \le & \stackrel{^}{\epsilon }\left(\stackrel{^}{h}\right)+\gamma \hfill \\ & \le & \stackrel{^}{\epsilon }\left({h}^{*}\right)+\gamma \hfill \\ & \le & \epsilon \left({h}^{*}\right)+2\gamma \hfill \end{array}$

The first line used the fact that $|\epsilon \left(\stackrel{^}{h}\right)-\stackrel{^}{\epsilon }\left(\stackrel{^}{h}\right)|\le \gamma$ (by our uniform convergence assumption). The second used the fact that $\stackrel{^}{h}$ was chosen to minimize $\stackrel{^}{\epsilon }\left(h\right)$ , and hence $\stackrel{^}{\epsilon }\left(\stackrel{^}{h}\right)\le \stackrel{^}{\epsilon }\left(h\right)$ for all $h$ , and in particular $\stackrel{^}{\epsilon }\left(\stackrel{^}{h}\right)\le \stackrel{^}{\epsilon }\left({h}^{*}\right)$ . The third line used the uniform convergence assumption again, to show that $\stackrel{^}{\epsilon }\left({h}^{*}\right)\le \epsilon \left({h}^{*}\right)+\gamma$ . So, what we've shown is the following: If uniform convergence occurs,then the generalization error of $\stackrel{^}{h}$ is at most $2\gamma$ worse than the best possible hypothesis in $\mathcal{H}$ !

where we get a research paper on Nano chemistry....?
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
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
how nano science is used for hydrophobicity
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
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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
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
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