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Review: pac bounds

Consider a finite collection of models F , and recall the basic PAC bound: for any δ > 0 , with probability at least 1 - δ

R ( f ) R ^ n ( f ) + log | F | + log ( 1 / δ ) 2 n , f F

where

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

and the loss is assumed to be bounded between 0 and 1. Note that we can write the inequality above as:

R ( f ) R ^ n ( f ) + log | F | δ 2 n

Letting δ f = δ | F | , we have:

R ( f ) R ^ n ( f ) + log ( 1 / δ f ) 2 n

This is precisely the form of Hoeffding's inequality, with δ f in place of the usual δ . In effect, in order to have Hoeffding's inequality hold with probability 1 - δ for all f F , we must distribute the “ δ -budget” or “confidence-budget” over all f F (in this case, evenly distributed):

f F δ f = f F δ | F | = δ

However, to apply the union bound, we do not need to distribute δ evenly among the candidate models. We only require:

f F δ f = δ

So, if p ( f ) are positive numbers satisfying f F p ( f ) = 1 , then we can take δ f = p ( f ) δ . This provides two advantages:

  1. By choosing p ( f ) larger for certain f , we can preferentially treat those candidates
  2. We do not need F to be finite and we only require f F p ( f ) = 1

Prefix codes are one way to achieve this. If we assign a binary prefix code of length c ( f ) to each f F , then the values p ( f ) = 2 - c ( f ) satisfy f F p ( f ) 1 according to the Kraft inequality.

The main point of this lecture is to examine how PAC bounds of the form w.p. 1 - δ

R ( f ) R ^ n ( f ) + c ( f ) log 2 + log ( 1 / δ ) 2 n , f F

can be used to select a model that comes close to achieving the best possible performace

inf f F R ( f )

Let f ^ n be the model selected from F using the training data { X i , Y i } i = 1 n . We will specify this model in a moment, but keep in mind that it is notnecessarily the model with minimum empirical risk as before. We would like to have

E [ R ( f ^ n ) ] - inf f F R ( f )

as small as possible. First, for any δ > 0 , define

f ^ n δ = arg min f F R ^ n ( f ) + C ( f , n , δ )

where

C ( f , n , δ ) c ( f ) log 2 + log ( 1 / δ ) 2 n

Then w.p. 1 - δ

R ( f ) R ^ n ( f ) + C ( f , n , δ ) , f F

and in particular,

R ( f ^ n δ ) R ^ n ( f ^ n δ ) + C ( f ^ n δ , n , δ ) ,

so, by the definition of f ^ n δ , f F

R ( f ^ n δ ) R ^ n ( f ) + C ( f , n , δ ) .

We will make use of the inequality above in a moment. First note that f F

E [ R ( f ^ n δ ) ] - R ( f ) = E [ R ( f ^ n δ ) - R ^ n ( f ) ] + E [ R ^ n ( f ) - R ( f ) ]

The second term is exactly 0, since E [ R ^ n ( f ) ] = R ( f ) .

Now consider the first term E [ R ( f ^ n δ ) - R ^ n ( f ) ] . Let Ω be the set of events on which

R ( f ^ n δ ) R ^ n ( f ) - C ( f , n , δ ) , f F

From the bound above, we know that P ( Ω ) 1 - δ . Thus,

E [ R ( f ^ n δ ) - R ^ n ( f ) ] = E [ R ( f ^ n δ ) - R ^ n ( f ) | Ω ] P ( Ω ) + E [ R ( f ^ n δ ) - R ^ n ( f ) | Ω c ] ( 1 - P ( Ω ) ) C ( f , n , δ ) + δ ( since 0 R , R ^ 1 , P ( Ω ) 1 and 1 - P ( Ω ) δ ) = c ( f ) log 2 + log ( 1 / δ ) 2 n + δ = c ( f ) log 2 + 1 2 log n 2 n + 1 n ( by setting δ = 1 n )

We can summarize our analysis with the following theorem.

Theorem

Complexity regularized model selection

Let F be a countable collection of models, and assign a positive number c ( f ) to each f F such that f F 2 - c ( f ) 1 . Define the minimum complexity regularized risk model

f ^ n = arg min f F R ^ n ( f ) + c ( f ) log 2 + 1 2 log n 2 n

Then,

E [ R ( f ^ n ) ] inf f F R ( f ) + c ( f ) log 2 + 1 2 log n 2 n + 1 n

This shows that

R ^ n ( f ) + c ( f ) log 2 + 1 2 log n 2 n

is a reasonable surrogate for

R ( f ) + c ( f ) log 2 + 1 2 log n 2 n

Histogram classifiers

Let X = [ 0 , 1 ] d be the input space and Y = { 0 , 1 } be the output space. Let F k , k = 1, 2, ...  denotes the collection of histogram classification rules withk equal volume bins. One choice of prefix code for this example is: k = 1 code = 0 , k = 3 code = 10 , k = 3 code = 110 and so on .... Then, if first code is corresponding to k f F k , followed by k = log 2 | F k | bits to indicate which of the 2 k histogram rules in F k is under consideration, we have

f F k c ( f ) = 2 k b i t s

Let f ^ n be the model that solves the minimization i.e.,

min k 1 min f F k R ^ n ( f ) + 2 k log 2 + 1 2 log n 2 n

That is, for each k, let

f ^ n ( k ) = arg min f F k R ^ n ( f )

Then select the best k according to

k ^ = arg min k 1 R ^ n ( f ^ n ( k ) ) + 2 k log 2 + 1 2 log n 2 n

and set

f ^ n = f ^ n ( k ^ )

Then,

E [ R ( f ^ n ) ] inf k 1 min f F k R ( f ) + 2 k log 2 + 1 2 log n 2 n + 1 n

It is a simple exercise to show that if d = 2 and the Bayes decision boundary is a 1-d curve, then by setting k = n and selecting the best f from F n we have

E [ R ( f ^ n ) ] = O ( n - 1 / 4 )
The complexity regularized classifier f ^ n adaptively achieves this rate, without user intervention.

Questions & Answers

Application of nanotechnology in medicine
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Bob
The nanotechnology is as new science, to scale nanometric
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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?
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Introduction about quantum dots in nanotechnology
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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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