Artificial Intelligence Statistical learning methods Chapter 20, AIMA 2 nd ed. Chapter 18, AIMA 3 rd...

Artificial Intelligence

Statistical learning methodsChapter 20, AIMA 2nd ed.Chapter 18, AIMA 3rd ed.

(only ANNs & SVMs)

Artificial neural networks

The brain is a pretty intelligent system.

Can we ”copy” it?

There are approx. 1011 neurons in the brain.

There are approx. 23109 neurons in the male cortex (females have about 15% less).

The simple model

• The McCulloch-Pitts model (1943)

Image from Neuroscience: Exploring the brain by Bear, Connors, and Paradiso

y = g(w0+w1x1+w2x2+w3x3)

w1

w2

w3

Neuron firing rate

Firi

ng r

ate

Depolarization

Transfer functions g(z)

The Heaviside function The logistic function

The simple perceptron

With {-1,+1} representation

Traditionally (early 60:s) trained with Perceptron learning.

0 if1

0 if1]sgn[)(

y

T

TT

xw

xwxwx

22110 xwxwwT xw

Perceptron learning

Repeat until no errors are made anymore1. Pick a random example [x(n),f(n)]

2. If the classification is correct, i.e. if y(x(n)) = f(n) , then do nothing

3. If the classification is wrong, then do the following update to the parameters (, the learning rate, is a small positive number)

Bn

Annf

class tobelongs )( if1

class tobelongs )( if1)(

x

xDesired output

)()( nxnfww iii

Example: Perceptron learning

The AND function

x1

x2x1 x2 f

0 0 -1

0 1 -1

1 0 -1

1 1 +1

Initial values:

= 0.3

1

1

5.0

w


The AND function

x1

x2x1 x2 f

0 0 -1

0 1 -1

1 0 -1

1 1 +1

1

1

5.0

w

This one is correctlyclassified, no action.


The AND function

x1

x2x1 x2 f

0 0 -1

0 1 -1

1 0 -1

1 1 +1

1

1

5.0

w

This one is incorrectlyclassified, learning action.

7.01

10

8.01

22

11

00

ww

ww

ww


The AND function

x1

x2x1 x2 f

0 0 -1

0 1 -1

1 0 -1

1 1 +1

7.0

1

8.0

w


7.01

10

8.01

22

11

00

ww

ww

ww


The AND function

x1

x2x1 x2 f

0 0 -1

0 1 -1

1 0 -1

1 1 +1

7.0

1

8.0

w

This one is correctlyclassified, no action.


The AND function

x1

x2x1 x2 f

0 0 -1

0 1 -1

1 0 -1

1 1 +1This one is incorrectlyclassified, learning action.

7.00

7.01

1.11

22

11

00

ww

ww

ww

7.0

1

8.0

w


The AND function

x1

x2x1 x2 f

0 0 -1

0 1 -1

1 0 -1

1 1 +1

7.0

7.0

1.1

w


7.00

7.01

1.11

22

11

00

ww

ww

ww


The AND function

x1

x2x1 x2 f

0 0 -1

0 1 -1

1 0 -1

1 1 +1

7.0

7.0

1.1

w Final solution

Perceptron learning

• Perceptron learning is guaranteed to find a solution in finite time, if a solution exists.

• Perceptron learning cannot be generalized to more complex networks.

• Better to use gradient descent – based on formulating an error and differentiable functions

N

n

nynfE1

2),()()( WW

Gradient search

)(WW EW

W

E(W)

“Go downhill”

The “learning rate” () is set heuristically

W(k)

W(k+1) = W(k) + W(k)

The Multilayer Perceptron (MLP)

• Combine several single layer perceptrons.

• Each single layer perceptron uses a sigmoid function (C)E.g.

x k

h j

h i

y l 1)exp(1)(

)tanh()(

zz

zz

outp

ut

inpu

t

Can be trained using gradient descent

Example: One hidden layer

• Can approximate any continuous function

(z) = sigmoid or linear, (z) = sigmoid.

x k

h j

y i

D

kkjkjj

J

jjijii

xwwh

hvvy

10

10

)(

)()(

x

xx

Training: Backpropagation(Gradient descent)

ijijij

N

nkjjk

N

njiij

w

tnhtvtntn

nxtntw

tnhtntv

tEt

]),([)(),(),(

where

)(),()(

]),([),()(

)()(

'

1

1

x

x

w

Support vector machines

Linear classifier on a linearly separable problem

There are infinitely manylines that have zero trainingerror.

Which line should we choose?



Choose the line with thelargest margin.

The “large margin classifier”

margin




Choose the line with thelargest margin.

The “large margin classifier”

margin


”Support vectors”

The plane separating and is defined by

The dashed planes are given by

margin

Computing the margin

w

aT xw

ba

baT

T

xw

xw

Divide by b

Define new w = w/b and = a/bmargin


w 1//

1//

bab

babT

T

xw

xw

1

1

xw

xwT

T

We have defined a scalefor w and b

We have

which gives

margin


margin

1)(

1

w

wxw

xw

T

T

w

x

x + w

w

2margin

w

Maximizing the margin isequal to minimizing

||w||

subject to the constraints

wTx(n) – +1 for all

wTx(n) – -1 for all

Quadratic programming problem, constraints can be included with Lagrange multipliers.


N

n

N

n

N

m

TmnnD mnmynyL

1 1 1

)()()()(2

1xx

Quadratic programming problem

N

n

Tnp nnyL

1

21)()(

2

1 xww

Minimize cost (Lagrangian)

Minimum of Lp occurs at the maximum of (the Wolfe dual)

Only scalar productin cost. IMPORTANT!

sn

Tn

T nnyy xxxwx )()(sgnsgn)(ˆ

Linear Support Vector Machine

Test phase, the predicted output

Where is determined e.g. by looking at one of the support vectors.

Still only scalar products in the expression.

How deal with nonlinear case?

• Project data into high-dimensional space Z. There we know that it will be linearly separable (due to VC dimension of linear classifier).

• We don’t even have to know the projection...!

)]([)( nn xz

Scalar product kernel trick

))(())(())(),(( mnmnK T xφxφxx

If we can find kernel such that

Then we don’t even have to know the mappingto solve the problem...

Valid kernels (Mercer’s theorem)

)](),([)]2(),([)]1(),([

)](),2([)]2(),2([)]1(),2([

)](),1([)]2(),1([)]1(),1([

NNKNKNK

NKKK

NKKK

xxxxxx

xxxxxx

xxxxxx

K

Define the matrix

If K is symmetric, K = KT, and positive semi-definite, thenK[x(i),x(j)] is a valid kernel.

Examples of kernels

dT jijiK

jijiK

)()()](),([

2/)()(exp)](),([2

xxxx

xxxx

First, Gaussian kernel. Second, polynomial kernel. With d=1 we have linear SVM.

Linear SVM often used with good success on highdimensional data (e.g. text classification).

Example: Robot color vision(Competition 1999)

Classify the Lego pieces into red, blue, and yellow.Classify white balls, black sideboard, and green carpet.

What the camera sees (RGB space)

Yellow

Green

Red

Mapping RGB (3D) to rgb (2D)

BGR

Bb

BGR

Gg

BGR

Rr

Lego in normalized rgb space

2Xx 6Cc

Input is 2D

x1

x2

Output is 6D:{red, blue, yellow,green, black, white}

MLP classifier

E_train = 0.21%E_test = 0.24%2-3-1 MLP

Levenberg-Marquardt

Training time(150 epochs):51 seconds

SVM classifier

E_train = 0.19%E_test = 0.20%SVM with

= 1000

2)(exp),( yxyx K

Training time:22 seconds

Machine Learning

• Machine learning (multilayer perceptrons, support vector machines, clustering) is covered in great detail in the course ”Learning Systems”.

Artificial Intelligence Statistical learning methods Chapter 20, AIMA 2 nd ed. Chapter 18, AIMA 3 rd...

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