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Take

Z = | R n ( f ) ^ - R ( f ) | and t = ϵ
P ( | R n ^ ( f ) - R ( f ) | ϵ ) E [ | R n ( f ) ^ - R ( f ) | 2 ] ϵ 2 var ( R ^ n ( f ) ) ϵ 2 = i = 1 n var ( L i n ) ϵ 2 = var ( ( X ) , Y ) n ϵ 2 = σ L 2 n ϵ 2 .

So, the probability goes to zero at a rate of at least n - 1 . However, it turns out that this is an extremely loose bound. Accordingto the Central Limit Theorem

R n ^ ( f ) = 1 n i = 1 n L i N R ( f ) , σ L 2 n as n

in distribution. This suggests that for large values of n,

P ( | R n ^ ( f ) - R ( f ) | ϵ ) O e - n ϵ 2 2 σ L 2 .

That is, the Gaussian tail probability is tending to zero exponentially fast.

Chernoff's bound

Note that for any nonnegative random variable Z and t > 0 ,

P ( Z t ) = P ( e s Z e s t ) E [ e s Z ] e s t , s > 0 by Markov's inequality .

Chernoff's bound is based on finding the value of s that minimizes the upper bound. If Z is a sum of independent random variables. For example, say

Z = i = 1 n ( f ( X i ) , Y i ) - R ( f ) = n R ^ n ( f ) - R ( f )

then the bound becomes

P i = 1 n ( L i - E [ L i ] ) t e - s t E [ e s i = 1 n ( L i - E [ L i ] ) ] e - s t i = 1 n E [ e s ( L i - E [ L i ] ) ] , from independence.

Thus, the problem of finding a tight bound boils down to finding a good bound for E [ s s ( L i - E [ L i ] ) ] . Chernoff ('52), first studied this situation for binary random variables. Then,Hoeffding ('63) derived a more general result for arbitrary bounded random variables.

Hoeffding's indequality

Theorem

Hoeffding's inequality

Let Z 1 , Z 2 , . . . , Z n be independent bounded random variables such that Z i [ a i , b i ] with probability 1. Let S n = i = 1 n Z i . Then for any t > 0 , we have

P ( | S n - E [ S n ] | t ) 2 e - 2 t 2 i = 1 n ( b i - a i ) 2 .

The key to proving Hoeffding's inequality is the following upper bound: if Z is a random variable with E [ Z ] = 0 and a Z b , then

E [ e s Z ] e s 2 ( b - a ) 2 8 .

This upper bound is derived as follows. By the convexity of theexponential function,

e s z z - a b - a e s b + b - z b - a e s a , for a z b .
Convexity of exponential function.

Thus,

E [ e s Z ] E Z - a b - a e s b + E b - Z b - a e s a = b b - a e s a - a b - a e s b , since E [ Z ] = 0 = ( 1 - θ + θ e s ( b - a ) ) e - θ s ( b - a ) , where θ = - a b - a .

Now let

u = s ( b - a ) and define φ ( u ) - θ u + log ( 1 - θ + θ e u ) .

Then we have

E [ e s Z ] ( 1 - θ + θ e s ( b - a ) ) e - θ s ( b - a ) = e φ ( u ) .

To minimize the upper bound let's express φ ( u ) in a Taylor's series with remainder :

φ ( u ) = φ ( 0 ) + u φ ' ( 0 ) + u 2 2 φ ' ' ( v ) for some v [ 0 , u ]
φ ' ( u ) = - θ + θ e u 1 - θ + θ e u φ ' ( u ) = 0 φ ' ' ( u ) = θ e u 1 - θ + θ e u - ( θ e u ) 2 ( 1 - θ + θ e u ) 2 = θ e u 1 - θ + θ e u ( 1 - θ e u 1 - θ + θ e u ) = ρ ( 1 - ρ ) .

Now, φ ' ' ( u ) is maximized by

ρ = θ e u 1 - θ + θ e u = 1 2 φ ' ' ( u ) 1 4 .

So,

φ ( u ) u 2 8 = s 2 ( b - a ) 2 8
E [ e s Z ] e s 2 ( b - a ) 2 8 .

Now, we can apply this upper bound to derive Hoeffding's inequality.

P ( S n - E [ S n ] t ) e - s t i = 1 n E [ e s ( L i - E [ L i ] ) ] e - s t i = 1 n e s 2 ( b i - a i ) 2 8 = e - s t e s 2 i = 1 n ( b i - a i ) 2 8 = e - 2 t 2 i = 1 n ( b i - a i ) 2 by choosing s = 4 t i = 1 n ( b i - a i ) 2

Similarly, P ( E [ S n ] - S n t ) e - 2 t 2 i = 1 n ( b i - a i ) 2 . This completes the proof of the Hoeffding's theorem.

Application

Let Z i = 1 f ( X i ) Y i - R ( f ) , as in the classification problem. Then for a fixed f, it follows fromHoeffding's inequality (i.e., Chernoff's bound in this special case) that

P ( | R n ^ ( f ) - R ( f ) | ϵ ) = P 1 n | S n - E [ S n ] | ϵ = P ( | S n - E [ S n ] | n ϵ ) 2 e - 2 ( n ϵ ) 2 n = 2 e - 2 n ϵ 2 .

Now, we want a bound like this to hold uniformly for all f F . Assume that F is a finite collection of models and let | F | denote its cardinality. We would like to bound the probability that max f F | R n ^ ( f ) - R ( f ) | ϵ . Note that the event

max f F | R n ^ ( f ) - R ( f ) | ϵ f F | R n ^ ( f ) - R ( f ) | ϵ .

Therefore

P max f F | R n ^ ( f ) - R ( f ) | ϵ = P f F | R n ^ ( f ) - R ( f ) | ϵ f F P ( | R n ^ ( f ) - R ( f ) | ϵ ) , the `` union of events '' bound 2 | F | e - 2 n ϵ 2 , by Hoeffding's inequality.

Thus, we have shown that with probability at least 1 - 2 | F | e - 2 n ϵ 2 , f F

| R n ^ ( f ) - R ( f ) | < ϵ .

And accordingly, we can be reasonably confident in selecting f from F based on the empirical risk function R ^ n .

Questions & Answers

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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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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.
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Porter
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
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