# 0.9 Error bounds in countably infinite spaces

 Page 1 / 2

## Introduction

In the last lecture , we studied bounds of the following form: for any $\delta >0$ , with probability at least $1-\delta$ ,

$R\left(f\right)\le {\stackrel{^}{R}}_{n}\left(f\right)+\sqrt{\frac{log|\mathcal{F}|+log\left(\frac{1}{\delta }\right)}{2n}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall f\in \mathcal{F}$

which led to upper bounds on the estimation error of the form

$E\left[R\left({\stackrel{^}{f}}_{n}\right)\right]-\underset{f\in F}{min}R\left(f\right)\phantom{\rule{4pt}{0ex}}\le \phantom{\rule{4pt}{0ex}}\sqrt{\frac{log|\mathcal{F}|+log\left(n\right)+2}{n}}.$

The key assumptions made in deriving the error bounds were:

• bounded loss function
• finite collection of candidate functions

The bounds are valid for every ${P}_{XY}$ and are called distribution-free.

## Deriving bounds for countably infinite spaces

In this lecture we will generalize the previous results in a powerful way by developing bounds applicable to possibly infinitecollections of candidates. To start let us suppose that $\mathcal{F}$ is a countable, possibly infinite, collection of candidate functions.Assign a positive number c( $f$ ) to each $f\in \mathcal{F}$ , such that

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}<\infty .$

The numbers c( $f$ ) can be interpreted as

• measures of complexity
• -log of prior probabilities
• codelengths

In particular, if P( $f$ ) is the prior probability of $f$ then

${e}^{-\left(-,log,p,\left(,f,\right)\right)}=p\left(f\right)$

so $c\left(f\right)\equiv -logp\left(f\right)$ produces

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}=\sum _{f\in \mathcal{F}}p\left(f\right)=1.$

Now recall Hoeffding's inequality. For each $f$ and every $ϵ>0$

$P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,ϵ\right)\le {e}^{-2n{ϵ}^{2}}$

or for every $\delta >0$

$P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta }\right)}{2n}}\right)\le \delta .$

Suppose $\delta >0$ is specified. Using the values c( $f$ ) for $f\in \mathcal{F}$ , define

$\delta \left(f\right)={e}^{-c\left(f\right)}\delta .$

Then we have

$P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)\le \delta \left(f\right).$

Furthermore we can apply the union bound as follows

$\begin{array}{ccc}\hfill P\left(\underset{f\in \mathcal{F}}{sup},\left\{R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),-,\sqrt{\frac{log\left(1/\delta \left(f\right)\right)}{2n}}\right\},\ge ,0\right)& \le & P\left(\bigcup _{f\in \mathcal{F}},R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)\hfill \\ & \le & \sum _{f\in \mathcal{F}}P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)\hfill \\ & \le & \sum _{f\in \mathcal{F}}\delta \left(f\right)=\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}\delta =\delta \hfill \end{array}.$

So for any $\delta >0$ with probability at least $1-\delta$ , we have that $\forall f\in \mathcal{F}$

$\begin{array}{ccc}\hfill R\left(f\right)& \le & {\stackrel{^}{R}}_{n}\left(f\right)+\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\hfill \\ & =& {\stackrel{^}{R}}_{n}\left(f\right)+\sqrt{\frac{c\left(f\right)+log\left(\frac{1}{\delta }\right)}{2n}}\hfill \end{array}.$

## Special case

Suppose $\mathcal{F}$ is finite and $c\left(f\right)=log|\mathcal{F}|\phantom{\rule{1.em}{0ex}}\forall f\in \mathcal{F}$ . Then

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}=\sum _{f\in \mathcal{F}}{e}^{-log|\mathcal{F}|}=\sum _{f\in \mathcal{F}}\frac{1}{|\mathcal{F}|}=1$

and

$\delta \left(f\right)=\frac{\delta }{|\mathcal{F}|}$

which implies that for any $\delta >0$ with probability at least $1-\delta$ , we have

$R\left(f\right)\le {\stackrel{^}{R}}_{n}\left(f\right)+\sqrt{\frac{log|\mathcal{F}|+log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall f\in \mathcal{F}.$

Note that this is precisely the bound we derived in the last lecture .

Choosing c( $f$ )

The generalized bounds allow us to handle countably infinite collections of candidate functions, but we require that

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}<\infty .$

Of course, if $c\left(f\right)=-logp\left(f\right)$ where $p\left(f\right)$ is a proper prior probability distribution then we have

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}=1.$

However, it may be difficult to design a probability distribution over an infinite class of candidates. The coding perspectiveprovides a very practical means to this end.

Assume that we have assigned a uniquely decodable binary code to each $f\in \mathcal{F}$ , and let c( $f$ ) denote the codelength for $f$ . That is, the code for $f$ is c( $f$ ) bits long. A very useful class of uniquely decodable codes are called prefix codes.

Prefix Code
A code is called a prefix codeif no codeword is a prefix of any other codeword.

## The kraft inequality

For any binary prefix code, the codeword lengths ${c}_{1}$ , ${c}_{2}$ , ... satisfy

$\sum _{i=1}^{\infty }{2}^{-{c}_{i}}\le 1.$

Conversely, given any ${c}_{1}$ , ${c}_{2}$ , ... satisfying the inequality above we can construct a prefix code with these codeword lengths.We will prove this result a bit later, but now let's see how this is useful in our learning problem.

Assume that we have assigned a binary prefix codeword to each $f\in \mathcal{F}$ , and let c( $f$ ) denote the bit-length of the codeword for $f$ . Set $\delta \left(f\right)={2}^{-c\left(f\right)}\delta$ . Then

$\begin{array}{ccc}\hfill P\left(\bigcup _{f\in \mathcal{F}},R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)& \le & \sum _{f\in \mathcal{F}}P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)\hfill \\ & \le & \sum _{f\in \mathcal{F}}\delta \left(f\right)=\sum _{f\in \mathcal{F}}{2}^{-c\left(f\right)}\delta =\delta \hfill \end{array}.$

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!