Ming-Feng Yeh 1

CHAPTER 10CHAPTER 10

Widrow-Hoff Widrow-Hoff LearningLearning

Ming-Feng Yeh 2

ObjectivesObjectives

Widrow-Hoff learning is an approximate steepest descent algorithm, in which the performance index is mean square error.

It is widely used today in many signal processing applications.

It is precursor to the backpropagation algorithm for multilayer networks.

Ming-Feng Yeh 3

ADALINE NetworkADALINE Network

ADALINE (Adaptive Linear Neuron) network and its learning rule, LMS (Least Mean Square) algorithm are proposed by Widrow and Marcian Hoff in 1960.Both ADALINE network and the perceptron suffer from the same inherent limitation: they can only solve linearly separable problems.The LMS algorithm minimizes mean square error (MSE), and therefore tires to move the decision boundaries as far from the training patterns as possible.

Ming-Feng Yeh 4

ADALINE NetworkADALINE Network

n = Wp + b a = purelin(Wp + b)

+

aS1

nS11 b

S1

W

SR

R

pR1

S

Single-layer perceptron

aS1+

nS11 b

S1

W

SR

R

pR1

S

Ming-Feng Yeh 5

Single ADALINESingle ADALINE

Set n = 0, then Wp + b = 0 specifies a decision boundary.The ADALINE can be used to classify objects into two categories if they are linearly separable.

11w

12w

1p

2p n a

1

b

nnpurelinabn

wwp

p

)(

, 12112

1

Wp

Wp

1p

2p

W0a

0a

Ming-Feng Yeh 6

Mean Square ErrorMean Square Error

The LMS algorithm is an example of supervised training.The LMS algorithm will adjust the weights and biases of the ADALINE in order to minimize the mean square error, where the error is the difference between the target output (tq) and the network output (pq).

1 ,1 pz

wx

b

])[(])[(][)( 2T22 zxx tEatEeEF

zxpw TT1 ba

E[·]: expected value

MSE:

Ming-Feng Yeh 7

Performance OptimizationPerformance Optimization

Develop algorithms to optimize a performance index F(x), where the word “optimize” will mean to find the value of x that minimizes F(x).

The optimization algorithms are iterative as

or : a search direction

: positive learning rate, which determines the length of the step

: initial guess

kkkk pxx 1 kkkk pxxx 1

kp

k

0x

Ming-Feng Yeh 8

Taylor Series ExpansionTaylor Series Expansion

Taylor series:

Vector case:

)()(

)()()()()(

)(

3)(!3

12)(!2

1)(3

3

2

2

xxxF

xxxxxxxFxF

xxdxxdF

xxdx

xFd

xxdx

xFd

xxdxxdF

)()()(

)()()()()()(

))(()())(()(

)()()()(

)()()()()()(

2!2

1

3311!21

2211!21

2!2

1211!2

1

11

31

2

21

2

2

2

21

2

1

xxxx

xxxxxxxxx

xx

xx

xxxx

xx

xxxx

xxxx

xxxx

xxxx

T

TTT

xxxx

nnxx

nnxx

FF

FFF

xxxxFxxxxF

xxFxxF

xxFxxFFF

n

n

),...,,( 21 nxxxx

Ming-Feng Yeh 9

Gradient & HessianGradient & Hessian

Gradient:

Hessian:

)...,,,()( 21 nxxxFF xT

n

Fx

Fx

Fx

F

)()()()(21

xxxx

)()()(

)()()(

)()()(

)(

2

2

2

2

1

2

2

2

22

2

12

21

2

21

2

21

2

2

xxx

xxx

xxx

x

Fx

Fxx

Fxx

Fxx

Fx

Fxx

Fxx

Fxx

Fx

F

nnn

n

n

Ming-Feng Yeh 10

Directional DerivativeDirectional Derivative

The ith element of the gradient, F(x)xi, is the first derivative of performance index F along the xi axis.

Let p be a vector in the direction along which we wish to know the derivative.Directional derivative:

. Find the derivative of F(x) at the point in the direction

pxp )(FT22

21 2)( xxF x

T5.05.0x T12 p

0

5

2

112

)(

2

1

4

2)(

2

1

p

xpx

xx

xx

F

x

xF

T

Ming-Feng Yeh 11

Steepest DescentSteepest Descent

Goal: The function F(x) can decrease at each iteration, i.e.,

Central idea: first-order Taylor series expansion

Any vector pk that satisfies is called a descent direction.

A vector that points in the steepest descent direction is

Steepest descent:

)()( 1 kk FF xx

kFFFF kk

Tkkkk xx

xgxgxxxx )(,)()()( 1

00)()( 1 kTkk

Tkkk

Tkkk FF pgpgxgxx

0kTk pg

kk gp

kkkkkkk gxpxx 1

kgx

Ming-Feng Yeh 12

Approximated-Based FormulationApproximated-Based Formulation

Given input/output training data: {p1,t1}, {p2,t2},…, {pQ,tQ}. The objective of network training is to find the optimal weights to minimize the error (minimum-squares error) between the target value and the actual response.

Model (network) function:

Least-squares-error function:

The weight vector x can be training by minimizing the error function along the gradient-descent direction:

),,( xza TTb 1 , pzwx 2),(),( xzxz tE

x

xzxz

x

xzx

),(),(

),( tE

Ming-Feng Yeh 13

Delta Learning RuleDelta Learning Rule

ADALINE:

Least-Squares-Error Criterion:

minimize

Gradient:

Delta learning rule:

zxxz TR

jjj bpwa ),(

T

Tb

1pz

wx

22

1atE

jjj

patw

a

a

E

w

E

atb

a

a

E

b

E

xx

E )()()1(

)()()1(

atkbkb

patkwkw jjj

Ming-Feng Yeh 14

Mean Square ErrorMean Square Error

matrixn correlatioinput :][ and between n vector correlatio-cross :][

][ 2

TEttE

tEc

zzRzzh

022)2()( TT RxhRxxhxx cF

hRx 1

xzzxzxxzzxzx

zxx

][][2][]2[

])[()(

TTT2

TTT2

2T

EtEtEttE

tEF

Rxxhx TT2 c

matrix symmetric: ,2)(

ectorconstant v: ,)()(TT

TT

RRxxRRxRxx

hhhxxh

Ming-Feng Yeh 15

Mean Square ErrorMean Square Error

If the correlation matrix R is positive definite, there will be a unique stationary point , which will be a strong minimum.Strong Minimum: the point is a strong minimum of F(x) if a scalar exists, such that

for all x such that .Global Minimum: the point is a unique global minimum of F(x) for all .Weak Minimum: the point is a weak minimum of F(x) if it is not a strong minimum, and a scalar exists, such that for all x such that .

0

hRx 1

x)()( xxx FF

0 xx

0xx

)()( xxx FF0 x

Ming-Feng Yeh 16

LMS AlgorithmLMS Algorithm

LMS algorithm is to locate the minimum point.

Use an approximate steepest descent algorithm to estimate the gradient.

Estimate the mean square error F(x) by

Estimated gradient:

)()()()(ˆ 22 kekaktF x

)()(ˆ 2 keF x

Rjw

keke

w

keke

jjj ,...,2,1for

)()(2

)()(

22

b

keke

b

keke R

)(

)(2)(

)(2

12

Ming-Feng Yeh 17

LMS AlgorithmLMS Algorithm

bkpw

kp

kpkp

wwwbkkaR

iii

R

R

1

2

1

21 )(

)(

)()(

)()( Wp

bkpwktkaktkeR

iii

1

)()()()()(

1)(

),()(

b

kekp

w

kej

j

)()(2)()(ˆ 2 kkekeF zx

Ming-Feng Yeh 18

LMS AlgorithmLMS Algorithm

The steepest descent algorithm with constant learning rate is

kFkk xx

xxx )(1

)()(2)()(ˆ 2 kkekeF zx )()(21 kkekk zxx

Matrix notation of LMS algorithm:

)(2)()1(

)()(2)()1( T

kkk

kkkk

ebb

peWW

The LMS algorithm is also referred to as the delta rule or the Widrow-Hoff learning algorithm.

Ming-Feng Yeh 19

Quadratic FunctionsQuadratic FunctionsGeneral form of quadratic function:

Ax

dAxx

Axxxdx

)(

)(

)(

2

T21T

G

G

cG

ADALINE network mean square error:Rxxhxx TT2)( cF

RAhd 2 ,2

(A: Hessian matrix)

If the eigenvalues of the Hessian matrix are all positive, then the quadratic function will have one unique global minimum.

Ming-Feng Yeh 20

Stable Learning RatesStable Learning RatesSuppose that the performance index is a quadratic function:

cG xdAxxx TT

21)(

dAxx )(G

Steepest descent algorithm with constant learning rate:

)(1 dAxxgxx kkkkk

dxAIx kk ][1

A linear dynamic system will be stable if the eigenvalues of the matrix [I-A] are less than one in magnitude.

Ming-Feng Yeh 21

Stable Learning RatesStable Learning RatesLet {1, 2,…, n} and {z1,z2,…, zn} be the eigenvalues and eigenvectors of the Hessian matrix. Then

iiiiiiii zzzAzzzAI )1(][

Condition for the stability of the steepest descent algorithm is then

11 i

Assume that the quadratic function has a strong minimum point, then its eigenvalues must be positive numbers. Hence,

i 2

This must be true for all eigenvalues:max

2

Ming-Feng Yeh 22

Analysis of ConvergenceAnalysis of ConvergenceIn the LMS algorithm , xk is a function only of z(k-1), z(k-2),…, z(0).

)()(21 kkekk zxx

Assume that successive input vectors are statistically independent, then xk is independent of z(k).

The expected value of the weight vector will converge to . This is the minimum MSE solution.

hRx 1 ]}[{ 2

keE,][1 dxAIx kk RAhd 2 ,2

hxRIx 2][]2[][ ssss EEThe condition on stability isThe steady state solution isor .

max10

xhRx 1][ ssE

Ming-Feng Yeh 23

Orange/Apple ExampleOrange/Apple Example

1,

111

,1,11

1

2211 tt pp

5.01

0.2,0.1,0.0

.101010101

][

max

T222

1T112

1T

ppppppER

In practical applications, the stable learning rate might NOT be practical to calculate R, and could be selected by trial and error.

Ming-Feng Yeh 24

Orange/Apple ExampleOrange/Apple ExampleStart, arbitrary, with all the weights set to zero, and then will apply input p1, p2, p1, p2, etc., in that order, calculating the new weights after each input is presented.

1)0()0()0()0(0)0()0()0()0( 11 atatea pWpW

1,

111

,1,11

1

2211 tt pp

2.0

000)0(

W

.4.04.04.0)0()0(2)0()1( T pWW e

4.1)1()1()1()1(4.0)1()1()1()1( 22 atatea pWpW

.16.096.016.0)1()1(2)1()2( T pWW e

Ming-Feng Yeh 25

Orange/Apple ExampleOrange/Apple Example

36.0)2()2()2()2(

64.0)2()2()2()2(

1

1

atate

a pWpW

.0160.01040.10160.0)2()2(2)2()3( T pWW e .010)( W

This decision boundary falls halfway between the two reference patterns. The perceptron rule did NOT produce such a boundary,

The perceptron rule stops as soon as the patterns are correctly classified, even though some patterns may be close to the boundaries. The LMS algorithm minimizes the mean square error.

Ming-Feng Yeh 26

Solved Problem P10.2Solved Problem P10.2Category I:

Category II:

TT 11,11 21 pp

T223 p

1p2p

3p

IIISince they are linear separable, we can design an ADALINE network to make such a distinction.As shown in figure, 1 ,1 ,3 1211 wwb

Category III:

Category IV:

TT 11,11 21 pp

T013 p1p

2p3pThey are NOT linear separable, so

an ADALINE network CANNOT distinguish between them.

Ming-Feng Yeh 27

Solved Problem P10.3Solved Problem P10.3

1,

11

,1,11

2211 tt pp

These patterns occur with equal probability, and they are used to train an ADALINE network with no bias. What does the MSE performance surface look like?

2122

212 ,212)( wwwwwcF TT xRxxhxx

1)1(5.015.0][ 222 tEc

10

11

)1(5.011

15.0][ zh tE

1001

5.05.0][ 2211TTTE ppppzzR

Ming-Feng Yeh 28

Solved Problem P10.3Solved Problem P10.3

10

10

1001

11hRx

2122

212 ,212)( wwwwwcF TT xRxxhxx

0 1 2 3-3 -2 -1

0

1

2

3

-2

-1

4

1w

2wThe Hessian matrix of F(x), 2R, has both eigenvalues at 2. So the contour of the performance surface will be circular. The center of the contours (the minimum point) is .x

Ming-Feng Yeh 29

Solved Problem P10.4Solved Problem P10.4

1,

11

,1,11

2211 tt pp

Train the network using the LMS algorithm, with the initial guess set to zero and a learning rate = 0.25.

21

21

)0()0(2)0()1(,101)0()0()0(

,011

00)0(

Teate

purelina

pWW

10)1()1(2)1()2(

,101)1()1()1(

,01

1)1( 2

121

Teate

purelina

pWW

w1 w1

Ming-Feng Yeh 30

Tapped Delay LineTapped Delay Line

D

D

D

)(ky )()(1 kykp

)1()(2 kykp

)1()( RkykpR

At the output of the tapped delay line we have an R-dim. vector, consisting of the input signal at the current time and at delays of from 1 to R–1 time steps.

Ming-Feng Yeh 31

Adaptive FilterAdaptive Filter

)(kn)(ka

1

b

11w

12wD

D

D

)(ky

Rw1

bikyw

bpurelinkaR

ii

1

1 )1(

)()( Wp

Ming-Feng Yeh 32

Solved Problem P10.1Solved Problem P10.1

)(kn )(ka

11w

12wD

)(ky

D13w

4)1(,5)0(,...}0,0,0,4,5,0,0,0{...,)}({

3 ,1 ,2 131211

yyky

www

Just prior to k = 0 ( k < 0 ):Three zeros have enteredthe filter, i.e., y(3) = y(2) = y(1) = 0, the output just prior to k = 0 is zero.

k = 0: 10005

312)0()0(

Wpa

Ming-Feng Yeh 33

Solved Problem P10.1Solved Problem P10.1

k = 1: 13054

312)1()1(

Wpa

k = 2: 1954

0312)2()2(

Wpa

k = 3: 124

00

312)3()3(

Wpa

k = 4: 0000

312)4()4(

Wpa

Ming-Feng Yeh 34

Solved Problem P10.1Solved Problem P10.1

The effect of y(0) last from k = 0 through k = 2, so it will have an influence for three time intervals.This corresponds to the length of the impulse response of this filter.

0)4(,12)3(,19)2(,13)1(,10)0(,0)1( aaaaaa

)2()1(

)()()( 131211

kyky

kywwwkka Wp

Ming-Feng Yeh 35

Solved Problem P10.6Solved Problem P10.6

)(kn )(ka

)(ky11w

12w

D

D +

+

)()( kykt

)(ke

Application of ADALINE: adaptive predictorThe purpose of this filter is to predict the next value of the input signal from the two previous values.Suppose that the input signal is a stationary random process with autocorrelation function given by

.1)2(,1)1(,3)0(,)()()( yyyy CCCnkykyEnC

Ming-Feng Yeh 36

Solved Problem P10.6Solved Problem P10.6Sketch the contour plot of the performance index (MSE).i.

)2()1(

)()(kyky

kk pz

.1)2(,1)1(,3)0(,)()()( yyyy CCCnkykyEnC

3)0()()( 22 yCkyEktEc

3113

)0()1()1()0(

yy

yyT

CCCC

E zzR

11

)2()1(

)2()()1()(

y

y

CC

kykykyky

EtE zh

Ming-Feng Yeh 37

Solved Problem P10.6Solved Problem P10.6Performance Index (MSE): Rxxhxx TT2)( cFThe optimal weights are

21

21

83

84

81

83

11

11

11

3113

hRx

The Hessian matrix is Eigenvalues: 1 = 4, 2 = 8.

Eigenvectors:

6226

2)(2 RAxF

11

,11

21 vv

The contours of F(x) will be elliptical, with the long axis of each ellipse along the 1st eigenvector, since the 1st eigenvalue has the smallest magnitude.The ellipses will be centered at .x

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

1v

2v

0 1 2-1-2

0

1

2

-1

-2

Ming-Feng Yeh 38

Solved Problem P10.6Solved Problem P10.6The maximum stable value of the learning for the LMS algorithm:

ii.25.0822 max

x

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2-1-2

0

1

2

-1

-2

The LMS algorithm is approximate steepest descent, so the trajectory for small learning rates will move perpendicular to the contour lines.

iii.

Ming-Feng Yeh 39

ApplicationsApplications

Noise cancellation system to remove 60-Hz noise from EEG signal (Fig. 10.6)

Echo cancellation system in long distance telephone lines (Fig. 10.10)

Filtering engine noise from pilot’s voice signal (Fig. P10.8)