5.3 Algorithmic Stability Bounds
Summarized by:
Sang Kyun Lee
Robustness 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)
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5.3.1 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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Notations

Given
&


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m-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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m-stability (2/2)

Proof (theorem 5.27)
l ( f z ( x), t )  l ( f zi z ( x), t )
= {l ( f z ( x), t )  l ( f z \i ( x), t )}  {l ( f z \i ( x), t )  l ( f zi z ( x), t )}
| l ( f z ( x), t )  l ( f zi z ( x), t ) |
 | l ( f z ( x), t )  l ( f z \i ( x), t ) |  | l ( f z \i ( x), t )  l ( f zi z ( x), t ) |
 2 m
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Lipschitz Loss Function (1/3)
 Thus, given Lipschitz continuous loss function l,
| l ( f z ( x), t )  l ( f ziz ( x), t ) |  Cl  | f z ( x)  f z \i ( x) |
 That is, we can use the difference of the two functions to bound
the losses incurred by themselves at any test object x.
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Lipschitz Loss Function (2/3)

Examples of Lipschitz continuous loss functions
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Lipschitz Loss Function (3/3)

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 Bound
for Regression Estimation

Now, in order to obtain generalization error bounds for mstable 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
Little bit crappy here!
which should preferably not depend on i 2 {1,…,m}
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Algorithmic Stability Bound
for Regression Estimation (C9.2 – 1/8)
Quick Proof:
Expectation over the random draw of training
samples z 2 Zm
=
=
=
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Algorithmic Stability Bound
for Regression Estimation (C9.2 – 2/8)
Quick Proof:
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Algorithmic Stability Bound
for Regression Estimation (C9.2 – 3/8)
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Algorithmic Stability Bound
for Regression Estimation (C9.2 – 4/8)

Proof
by Lemma C.21
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Algorithmic Stability Bound
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(-2/m m2))

 This result is slightly surprising, because
VC theory indicates that the training error Remp is only a good indicator of
the 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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Algorithmic Stability Bound
for Regression Estimation (C9.2 – 6/8)

However, recall that
 VC theory is used for empirical risk minimization algos which
only 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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Algorithmic Stability Bound
for Regression Estimation (C9.2 – 7/8)

Let’s consider m-stable algorithm A s.t. m · m-1
 From thm 5.32,
!
with probability of at least 1-.
is an amazingly tight generalization error bound whenever 
¿ m 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),
 This
regardless
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From new error bounds expression,
2
2(4  b) 2 ln( 1 )
R[ A( z )]  Remp [ A( z ), z ] 

.
m
m
Since we assume  -insensitive loss,
1 m
2
2(4  b) 2 ln( 1 )
R[ A( z )]  EZ [  max(| ti  ti |  , 0)] 

m i 1
m
m
1 m
 EZ [  max(| ti  w, xi |  , 0)]  ...
m i 1
1 m
 EZ [  (  i   )]  ...
m i 1
1 m
  i  ...
m i 1
1 T
2
2(4  b) 2 ln( 1 )
 ξ 1

m
m
m
Cl 2
(Cl  1,  
)
2
1  T 2 
2(4 2  1  b) 2 ln( 1 )
 ξ 1  
m
 
m
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5.3.2 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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m 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)

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)

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
: llin (tˆ, y)  max{1  ytˆ, 0} 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)=0
1.
2.
Thus the steepness of the function is 1/
Therefore the Lipschitz constant is also 1/
3. The function should be in the interval [0,1] because the zeroone loss will never exceed 1.
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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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Algorithmic Stability for Classification (2/3)
From the error bounds for classification,

Consider this thm for the
special case of linear soft
margin SVM for classification
(see 2.4.2)
 WLOG, assume  = 1
2
R[ sign( A( z ))]  R emp [ A( z ), z ] 
 m 2

2
2(2 2  1) 2 ln( 1 )


.
m
Since we assume soft-margin loss,
R[ A( z )]  EZ [
1 m
 max(|1  yti |, 0)]  ...
m i 1
 EZ [
1 m
 max(|1  yi w, xi |, 0)]  ...
m i 1
 EZ [
1 m
 i )]  ...
m i 1

1 m
 i  ...
m i 1
1 T
2
 ξ 1
m
 m 2
2
2(2 2  1) 2 ln( 1 )


m
( =1)
1 T
1
( 1  1) 2 ln( 1 )
 ξ 1   2
m

m
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Algorithmic Stability for Classification (3/3)

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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Algorithmic Stability for Classification (4/4)
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