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 $\mathcal{H}$ !

Questions & Answers

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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
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?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Read also:

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Machine learning' conversation and receive update notifications?

 By