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SVM and Kernel method (nips tutorial)

SVM,

kernel

method,

nips

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SVM and Kernel method (nips tutorial)

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SVM and Kernel Metho ds

Bernhard Scholkopf

Max-Planck-Institut f ur biologische Kybernetik

72076 T ubingen, Germany

Biowulf Technologies

305 Broadway, New York, NY 10007, USA

bernhard.schoelkopf@tuebingen.mpg.de

(these slides are available from www.kernel-machines.org)

B. Scholkopf, NIPS, 3 December 2001

Roadmap

1. ideas of statistical learning theory

2. kernels and feature spaces

3. Supp ort vector algorithms

B. Scholkopf, NIPS, 3 December 2001

Statistical Learning Theory

1. started by Vapnik and Chervonenkis in the Sixties

2. mo del: we observe data generated by an unknown sto chastic

regularity

learning

= extraction of the regularity from the data

4. the analysis of the learning problem leads to notions of

capacity

of the function classes that a learning machine can implement.

support vector machines

use a particular typ e of function class:

classiers with large \margins" in a feature space induced by a

kernel

[49, 50]

B. Scholkopf, NIPS, 3 December 2001

Example: Regression Estimation



Data:

input-output pairs (

; y

)





Regularity:

(

; y

)

; : : :

(

; y

) drawn from P(

x; y

)



Learning:

cho ose a function

such that the error,

averaged over P, is minimized.



Problem:

P is unknown, so the average cannot b e computed

| need an \

induction

principle"

B. Scholkopf, NIPS, 3 December 2001

Pattern Recognition

Learn

X ! f

from examples

(

; y

)

; : : : ;

(

; y

)

2 X f

;

generated i.i.d. from P(

x; y

)

;

such that the exp ected misclassication error on a test set, also

drawn from P(

x; y

[

] =

(

)



)

x; y

)

;

is minimal

(Risk Minimization (RM))

Problem

: P is unknown.

!

need an

induction principle

Empirical risk minimization (ERM):

replace the average over

x; y

) by an average over the training sample, i.e.

minimize the

training error

emp

[

] =

(

)



B. Scholkopf, NIPS, 3 December 2001

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