Lecture ANN 4

4

4

Learning Rules

p₁ t₁

{ , }, {p₂, }t₂ , … , {p_Q,t_Q}

•

Supervised Learning

Network is provided with a set of examples of proper network behavior (inputs/targets)

•

Reinforcement Learning

Network is only provided with a grade, or score, which indicates network performance

•

Unsupervised Learning

4

Perceptron Architecture

p a 1 n

A

W

A

A

b

Rx1

SxR

Sx1

Sx1

Sx1

Input

R

AA

S

AA

AA

a = hardlim(Wp+b)

Hard Limit Layer _W

w_{1 1,} w_{1 2,} … w_{1, R}

w_{2 1,} w_{2 2,} … w_{2, R}

w_{S, 1} w_S,2 … w_{S R,} =

w

i

w_{i 1,}

w_{i 2,}

w_{i, R}

= W wT 1 wT 2 wT S =

4

Single-Neuron Perceptron

p₁

a n

Inputs

b

p_{2 w}

1,2

w_1,1

1

AA

AA

Σ

AA

AA

a = hardlim(Wp+b)

Two-Input Neuron

w_{1 1,} = 1 w_{1 2,} = 1 _{b = –1}

p₁

p₂

1wTp+b=0

a = 1

a = 0

1

1

4

Decision Boundary

1w ₁w

wT

1 p + b = 0 w

T

1 p = –b

• All points on the decision boundary have the same inner product with the weight vector.

• Therefore they have the same projection onto the weight vector, and they must lie on a line orthogonal to the

weight vector

1wTp+b=0

4

Example - OR

p₁ 0

0

= , t₁ = 0

 

 

 

p₂ 0

1

= , t₂ = 1

 

 

 

p₃ 1

0

= , t₃ = 1

 

 

 

p₄ 1

1

= , t₄ = 1

 

 

4

OR Solution

1w

OR

w

1 0.5

0.5 =

wT

1 p + b 0.5 0.5 0 0.5

b

+ 0.25 + b 0

= = = ⇒ b = –0.25

Weight vector should be orthogonal to the decision boundary.

4

Multiple-Neuron Perceptron

Each neuron will have its own decision boundary.

wT

i p + b_i = 0

A single neuron can classify input vectors

into two categories.

4

Learning Rule Test Problem

p₁ t₁

{ , }, {p₂,t₂} , ,… {p_Q,t_Q}

p₁ 1

2

= , t₁ = 1

 

 

 

p₂ –1

2

= , t₂ = 0

 

 

 

p₃ 0

1 –

= , t₃ = 0

      p₁ a n Inputs

p_{2 w}

1,2 w_1,1

AA

AA

Σ

AA

AA

a = hardlim(Wp)

4

Starting Point

1w

1

3 2

w

1 1.0

0.8 – =

Present p₁ to the network:

a hardlim(₁wTp₁) hardlim _{1.0 –0.8} 1

2

 

 

 

= =

4

Tentative Learning Rule

1w

1 2

• Set

₁

w

to

p

1 – Not stable

• Add

p

₁

to

₁

w

If t = 1 and a = 0, then ₁wnew = ₁wold + p

w

1

new

w

1

old

p₁

+ 1.0

0.8 –

1 2

+ 2.0

1.2

= = =

4

Second Input Vector

1 2

If t = 0 and a = 1, then ₁wne w = ₁wold – p

a hardlim(₁wTp₂) hardlim _{2.0 1.2} –1

2

 

 

 

= =

a = hardlim(0.4) = 1 (Incorrect Classification)

Modification to Rule:

w

1

new

w

1

old

p₂

– 2.0

1.2

1 –

2

– 3.0

0.8 –

4

Third Input Vector

1w

1

3 2

Patterns are now correctly classified.

a hardlim(₁wTp₃) hardlim _{3.0 –0.8} 0

1 –       = =

a = hardlim 0.8( ) = 1 (Incorrect Classification)

w 1 new w 1 old p₃ – 3.0 0.8 – 0 1 – – 3.0 0.2 = = =

4

Unified Learning Rule

If t = 1 and a = 0, then ₁wnew = ₁wold + p

If t = 0 and a = 1, then ₁wnew = ₁wold – p

If t = a, then ₁wnew = ₁wold

e = t – a

If e = 1, then ₁wnew = ₁wold + p

If e = –1, then ₁wnew = ₁wold – p

If e = 0, then ₁wnew = ₁wold

4

Multiple-Neuron Perceptrons

w

i

ne w

w

i

old

e_ip

+ =

b_inew = b_iold + e_i

Wnew = Wold + epT

bnew = bold + e

To update the ith row of the weight matrix:

4

Apple/Banana Example

W = _{0.5 –1 –0.5} b = 0.5

a hardlim(Wp₁ + b) hardlim _{0.5 –1 –0.5}

1 – 1 1 – 0.5 +           = = Training Set Initial Weights First Iteration p₁ 1 – 1 1 – t₁ , ₁ = =           p₂ 1 1 1 – t₂ , ₀ = =          

4

Second Iteration

a hardlim (Wp₂ + b) hardlim _{–0.5 0 –1.5}

1 1 1 –

1.5

( )

+

( )

= =

a = hardlim (2.5) = 1

e = t₂ – a = 0 – 1 = –1

Wne w = Wold + epT = _{–0.5 0 –1.5} + ( )–1 _{1 1 –1} = _{–1.5 –1 –0.5}

4

Check

a hardlim (Wp₁ + b) hardlim _{–1.5 –1 –0.5}

1 –

1 1 –

0.5 +

( )

= =

a = hardlim (1.5) = 1 = t₁

a hardlim (Wp₂ + b) hardlim _{–1.5 –1 –0.5}

1 1 1 –

0.5 +

( )

= =

4

Perceptron Rule Capability

The perceptron rule will always

converge to weights which accomplish

the desired classification, assuming that

4

Perceptron Limitations

wT

1 p + b = 0

Linear Decision Boundary