5.3 algorithmic stability bounds

5.3 Algorithmic Stability Bounds5.3 Algorithmic Stability Bounds

Summarized by:

Sang Kyun Lee

(c) 2005 SNU CSE Biointelligence Lab

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Robustness of a learning algorithmRobustness of a learning algorithm

Instead of compression and reconstruction function, now we think about the “robustness of a learning algorithm A”

Robustness a measure of the influence of an additional training example (x, y) 2 Z

on the learned hypothesis A(z) 2 H quantified in terms of the loss achieved at any test object x 2 X A robust learning algorithm guarantees

|expected risk - empirical risk| < M

even if we replace one training example by its worst counterpart This fact is of great help when using McDiarmid’s inequality (A.119)

– a large deviation result perfectly suited for the current purpose


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McDiarmid’s Inequality (A.119)McDiarmid’s Inequality (A.119)


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5.3.1 Algorithmic Stability for Regression5.3.1 Algorithmic Stability for Regression

Framework Training sample:

drawn iid from an unknown distribution

Hypothesis: a real-valued function

Loss function: l : R £ R ! R a function of predicted value and observed value t


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NotationsNotations

Given &


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mm-stability (1/2)-stability (1/2)

this implies robustness in the more usual sense of measuring the influence of an extra training example. This is formally expressed in the following theorem.


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mm-stability (2/2)-stability (2/2)

Proof (theorem 5.27)

\ \

\ \

( ( ), ) ( ( ), )

= { ( ( ), ) ( ( ), )} { ( ( ), ) ( ( ), )}

| ( ( ), ) ( ( ), ) |

| ( ( ), ) ( ( ), ) | | ( ( ), ) ( ( ), ) |

i z

i z

i z

i z

z z

z z i z i z

z z

z z i z i z

l f x t l f x t

l f x t l f x t l f x t l f x t

l f x t l f x t

l f x t l f x t l f x t l f x t

m 2


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Lipschitz Loss Function (1/3)Lipschitz Loss Function (1/3)

Thus, given Lipschitz continuous loss function l,

That is, we can use the difference of the two functions to bound the losses incurred by themselves at any test object x.

\| ( ( ), ) ( ( ), ) | | ( ) ( ) |i zz z l z z il f x t l f x t C f x f x


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Examples of Lipschitz continuous loss functions


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Using the concept of Lipschitz continuous loss functinos we can upper bound the value of m for a large class of learning algorithms, using the following theorem (Proof at Appendix C9.1):

Using this, we’re able to cast most of the learning algorithms presented in Part I of this book into this framework


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Algorithmic Stability BoundAlgorithmic Stability Boundfor Regression Estimationfor Regression Estimation Now, in order to obtain generalization error bounds for m-st

able learning algorithms A we proceed as follows:1. To use McDiarmid’s inequality, define a random variable g(Z) which

measure |R[fz] – Remp[fz,z]| or |R[fz] – Rloo[A,z]|.(ex) g(Z) = R[fz] – Remp[fz,z]

2. Then we need to upper bound E[g] over the random draw of training samples z 2 Zm. This is because we’re only interested in the prob. that g(Z) will be larger than some prespecified .

3. We also need an upper bound on

which should preferably not depend on i 2 {1,…,m}

Little bit crappy here!


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Algorithmic Stability BoundAlgorithmic Stability Boundfor Regression Estimation (C9.2 – 1/8)for Regression Estimation (C9.2 – 1/8)

=

=

=

Quick Proof:Expectation over the random draw of training samples z 2 Zm


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Quick Proof:


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Proof

by Lemma C.21


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Algorithmic Stability BoundAlgorithmic Stability Boundfor Regression Estimation (C9.2 – 5/8)for Regression Estimation (C9.2 – 5/8) Summary:

The two bounds are essentially the same the additive correction ¼ m

the decay of the prob. is O(exp(-/m m))

This result is slightly surprising, because VC theory indicates that the training error Remp is only a good indicator of t

he generalization error when the hypothesis space has a small VC dimension (Thm. 4.7)

In contrast, the loo error disregards VC dim and is an almost unbiased estimator of the expected generalization error of an algorithm (Thm 2.36)


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However, recall that VC theory is used for empirical risk minimization algos which o

nly consider the training error as the coast function to be minimized

In contrast, in the current formulation we have to guarantee a certain stability of the learning algorithm

: in case of ! 0 (the learning algorithm minimizes the emp risk only, we can no longer guarantee a finite stability.


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Let’s consider m-stable algorithm A s.t. m · m-1

From thm 5.32,

! with probability of at least 1-.

This is an amazingly tight generalization error bound whenever ¿ because the expression is dominated by the second term

Moreover, this provides us practical guides on the possible values of the trade-off parameter . From (5.19),

m

regardless of the empirical term Remp[A(z),z]


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2 1

2 1

1

1

From new error bounds expression,

2 2(4 ) ln( )[ ( )] [ ( ), ] .

Since we assume -insensitive loss,

1 2 2(4 ) ln( )[ ( )] [ max(| | ,0)]

1[ max(| , | ,0)] ..

emp

m

Z i ii

m

Z i ii

bR A z R A z z

m m

bR A z E t t

m m m

E t w xm

1

1

22 1

2 2 1 2 1

.

1[ ( )] ...

1...

1 2 2(4 ) ln( ) ( 1, )

2

1 2(4 ) ln( )

m

Z ii

m

ii

ll

Em

m

CbC

m m m

b

m m

T

T

ξ 1

ξ 1


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5.3.2 Algorithmic Stability for Classification5.3.2 Algorithmic Stability for Classification

Framework Training sample:

Hypothesis:

Loss function: Confine to zero-one loss, although the following also applies to any loss that takes a finite

set of values.


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mm stability stability

For a given classification algorithm However, here we have m 2 {0,1} only.

m= 0 occurs if, for all training samples z 2 Zm and all test examples (x,y) 2 Z,

which is only possible if H only contains on hypothesis.

If we exclude this trivial case, then thm 5.32 gives trivial result


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Refined Loss Function (1/2)Refined Loss Function (1/2)

In order to circumvent this problem, we think about the real-valued output f(x) and the classifier of the form h(¢)=sign(f(¢)). As our ultimate interest is the generalization error

, Consider a loss function:

which is a upper bound of the function Advantage from this loss function settings:


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Refined Loss Function (2/2)Refined Loss Function (2/2)

Another useful requirement on the refined loss function l is Lipschitz continuity with a small Lipschitz constant This can be done by adjusting the linear soft margin loss

: where y 2 {-1,+1}

1. Modify this function to output at least on the correct side

2. Loss function has to pass through 1 for f(x)=01. Thus the steepness of the function is 1/2. Therefore the Lipschitz constant is also 1/

3. The function should be in the interval [0,1] because the zero-one loss will never exceed 1.

ˆ ˆ( , ) max{1 ,0}linl t y yt


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Algorithmic Stability for Classification (1/3)Algorithmic Stability for Classification (1/3)

• For ! 1, the first term is provably non-increasing whereas the second term is always decreasing


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Consider this thm for the special case of linear soft margin SVM for classification (see 2.4.2)

WLOG, assume = 1

22 1

2 2

2

1

1

From the error bounds for classification,

2(2 1) ln( )[ ( ( ))] [ ( ), ] .

Since we assume soft-margin loss,

1[ ( )] [ max(|1 |,0)] ...

1[ max(|1 , |,0)] ..

emp

m

Z ii

m

Z i ii

R sign A z R A z zm m

R A z E ytm

E y w xm

1

1

22 1

2 2

2

1 2 1

.

1[ )] ...

1...

2(2 1) ln( )1 ( =1)

1 1 ( 1) ln( )2

m

Z ii

m

ii

Em

m

m m m

m m

T

T

ξ 1

ξ 1


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This bounds provides an interesting model selection criterion, by which we select the value of (the assumed noise level).

In contrast to the result of Subsection 4.4.3, this bound only holds for the linear soft margin SVM

The results in this section are so recent that no empirical studies have yet been carried out


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5.3 algorithmic stability bounds

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