1 back-propagation. 2 objectives a generalization of the lms algorithm, called backpropagation, can...

Back-PropagationBack-Propagation

Objectives

A generalization of the LMS algorithm, called backpropagation, can be used to train multilayer networks.

Backpropagation is an approximate steepest descent algorithm, in which the performance index is mean square error.

In order to calculate the derivatives, we need to use the chain rule of calculus.

Motivation

The perceptron learning and the LMS algorithm were designed to train single-layer perceptron-like networks.

They are only able to solve linearly separable classification problems.Parallel Distributed ProcessingThe multilayer perceptron, trained by the backpropagation algorithm, is

currently the most widely used neural network.

Three-Layer Network

321 SSSR Number of neurons in each layer:R: number of inputs Sn: number of neurons in n layer

Pattern Classification: XOR gate

The limitations of the single-layer perceptron (Minsky & Papert, 1969)

011 tp

022 tp

133 tp

144 tp

Two-Layer XOR Network

Two-layer, 1-2-1 network

Individual Decisions

Solved Problem P11.1

Design a multilayer network to distinguish these categories.

T1 1111 p

T2 1111 p

T3 1111 p

T4 1111 p

Class I Class II01 bWp02 bWp

03 bWp04 bWp

There is no hyperplane that can separate these two categories.

Solution of Problem P11.1

Function Approximation

Two-layer, 1-2-1 network

1)( 21

.10,10,10,10 12

11 bbww

.0,1,1 221

21 bww

The centers of the steps occur where the net input to a neuron in the first layer is zero.

The steepness of each step can be adjusted by changing the network weights.

110100

110)10(012

wbpbpwn

Effect of Parameter Changes

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

20 15 10 5 0

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

Two-layer networks, with sigmoid transfer functions in the hidden layer and linear transfer functions in the output layer, can approximate virtually any function of interest to any degree accuracy, provided sufficiently many hidden units are available.

Backpropagation Algorithm

For multilayer networks the outputs of one layer becomes the input to the following layer.

1,...,2,1,0 ),( 1111 Mmmmmmm baWfaMaapa ,0

Performance Index

Training Set:

Mean Square Error:

Vector Case:

Approximate Mean Square Error:

Approximate Steepest Descent Algorithm

p1 t1{ , } p2 t2{ , } pQ tQ{ , }

F x E e2 = E t a– 2 =

F x E eTe = E t a–

Tt a– =

F̂ x t k a k – T t k a k – eTk e k = =

w i jm

k 1+ wi jm

w i jm

------------–= bim

k 1+ bim

---------–=

Chain Rule

If f(n) = en and n = 2w, so that f(n(w)) = e2w.

Approximate mean square error:

wndf )()())((

2)()())((

)()()]()([)]()([)(ˆ TT kkkkkkF eeatatx

ˆ)()1(

Sensitivity & Gradient

The net input to the ith neurons of layer m:

The sensitivity of to changes in the ith element of the net input at layer m:

Gradient:

mi bawn

1, 1 ,1

mi nFs ˆ

Steepest Descent Algorithm

The steepest descent algorithm for the approximate mean square error:

Matrix form:

,,, )(

ˆ)()1(

mji askw

mi skb

k 1+ Wm

am 1–

bmk 1+ bm

k sm–=

----------

---------

-----------

BP the Sensitivity

Backpropagation: a recurrence relationship in which the sensitivity at layer m is computed from the sensitivity at layer m+1.

Jacobian matrix:

Matrix RepressionThe i,j element of Jacobian matrix

Recurrence Relation

The recurrence relation for the sensitivity

The sensitivities are propagated backward through the network from the last layer to the first layer.

ˆ))((

.121 ssss MM

Backpropagation Algorithm

At the final layer:

)()()(ˆ

)()( M

)()(2 Mi

Mi ngats

))((2 atnFs MMM

Summary

The first step is to propagate the input forward through the network:

The second step is to propagate the sensitivities backward through the network:Output layer:Hidden layer:

The final step is to update the weights and biases:

1,...,2,1,0 ),( 1111 Mmmmmmm baWfapa 0

))((2 atnFs MMM 1,2,...,1 ,))(( 1T1 Mmmmmmm sWnFs

T1)()()1( mmmm kk asWW mmm kk sbb )()1(

BP Neural Network

jS mw,1

mjw ,1

mjiw ,

Layer m

Layer m-1

mw 1,1

miw 1,

S mw1,1

SS mmw,1

S mw,1

Si mw,

Layer MMa1

Layer 1

,1 RSw

Ex: Function Approximation

g p 14---p sin+=

1-2-1Network

Network Architecture

1-2-1Network

G(p) = 1+sin(/4 * p) for -2<=p <=2

Initial Values

W10 0.27–

0.41–= b1

0 0.48–

0.13–= W2

0 0.09 0.17–= b20 0.48=

Network ResponseSine Wave

-2 -1 0 1 2-1

Initial Network Response:

Forward Propagation

p 1= =

a1 f1 W1a0 b1+ l ogsig 0.27–

0.41–1

0.48–

0.13–+

logsig 0.75–

0.54–

f2 W2a1 b2

+ purelin 0.09 0.17–0.321

0.3680.48+( ) 0.446= = =

e t a– 1 4---p sin+

a2– 1 4---1 sin+

0.446– 1.261= = = =

1 e0.75+--------------------

1 e0.54+--------------------

0.368= =

Initial input:Output of the 1st layer:

Output of the 2nd layer:

error:

Transfer Func. Derivatives

))(1(1

1)()(2 ndn

Backpropagation

The second layer sensitivity:

The first layer sensitivity:522.2261.112

)]([2))((2 22222

enfn atFs

0997.0

0495.0

522.217.0

368.0)368.01(0

0321.0)321.01(

))(1(0

0))(1())(( 2

1122111 ssWnFs

Weight Update

Learning rate 1.0

0772.0171.0

368.0321.0]522.2[1.017.009.0

)()0()1( 1222

]732.0[]522.2[1.0]48.0[)0()1( 222 sbb

265.0]1[

0997.0

0495.01.0

)()0()1( 0111 TasWW

0997.0

0495.01.0

48.0)0()1( 111 sbb

The network transfer functions are 21 )()( nnf , n

and the input / target pair is given to be ))1(),1(( tp .

Perform one iteration of backpropagation with 1 .

For the network shown in the figure below, the initial weights and biases are chosen

to be 1)0(,1)0(,2)0(,1)0( 2211 bwbw .

Choice of Network Structure

Multilayer networks can be used to approximate almost any function, if we have enough neurons in the hidden layers.

We cannot say, in general, how many layers or how many neurons are necessary for adequate performance.

Illustrated Example 1

g p 1i 4----- p sin+=

-2 -1 0 1 2-1

1-3-1 Network 1i 2i

Illustrated Example 2

g p 164

------ p sin+=

-2 -1 0 1 2-1

1-2-1 1-3-1

Convergence

g p 1 p sin+=

-2 -1 0 1 2-1

Convergence to Global Min. Convergence to Local Min.The numbers to each curve indicate the sequence of iterations.

Generalization

In most cases the multilayer network is trained with a finite number of examples of proper network behavior:

This training set is normally representative of a much larger class of possible input/output pairs.

Can the network successfully generalize what it has learned to the total population?p1 t1{ , } p2 t2{ , } pQ tQ{ , }

Generalization Example

g p 14---p sin+= p 2– 1.6– 1.2– 1.6 2 =

-2 -1 0 1 2-1

1-2-1 1-9-1

For a network to be able to generalize, it should have fewer parameters than there are data points in the training set.

Generalize well Not generalize well

Objectives

The neural networks, trained in a supervised manner, require a target signal to define correct network behavior.

The unsupervised learning rules give networks the ability to learn associations between patterns that occur together frequently.

Associative learning allows networks to perform useful tasks such as pattern recognition (instar) and recall (outstar).

What is an Association?

An association is any link between a system’s input and output such that when a pattern A is presented to the system it will respond with pattern B.

When two patterns are link by an association, the input pattern is referred to as the stimulus and the output pattern is to referred to as the response.

Classic Experiment

Ivan Pavlov He trained a dog to salivate at the sound of a bell, by ringing the bell

whenever food was presented. When the bell is repeatedly paired with the food, the dog is conditioned to salivate at the sound of the bell, even when no food is present.

B. F. Skinner He trained a rat to press a bar in order to obtain a food pellet.

Associative Learning

Anderson and Kohonen independently developed the linear associator in the late 1960s and early 1970s.

Grossberg introduced nonlinear continuous-time associative networks during the same time period.

Simple Associative Network

Single-Input Hard Limit AssociatorRestrict the value of p to be either 0 or 1, indicating whether a stimulus is

absent or present.

The output a indicates the presence or absence of the network’s response.

stimulus. no ,0

stimulus. ,1p

response. no ,0response. ,1

)5.0()( wphardlimbwphardlima

Two Types of Inputs

Unconditioned Stimulus

Analogous to the food presented to the dog in Pavlov’s experiment.

Conditioned Stimulus

Analogous to the bell in Pavlov’s experiment.

The dog salivates only when food is presented. This is an innate that does not have to be learned.

Banana Associator

An unconditioned stimulus (banana shape) and a conditioned stimulus (banana smell)

The network is to associate the shape of a banana, but not the smell.

n a0p 0w5.0 ,0 ,10 bww

detected.not shape ,0

detected. shape ,10p

detected.not smell ,0

detected. smell ,1p )( 00 bwppwhardlima

Associative Learning

Both animals and humans tend to associate things occur simultaneously.If a banana smell stimulus occurs simultaneously with a banana concept

response (activated by some other stimulus such as the sight of a banana shape), the network should strengthen the connection between them so that later it can activate its banana concept in response to the banana smell alone.

Unsupervised Hebb Rule

Increasing the weighting wij between a neuron’s input pj and output ai in proportion to their product:

Hebb rule uses only signals available within the layer containing the weighting being updated. Local learning rule

Vector form:

Learning is performed in response to the training sequence

)()()1()( qpqaqwqw jiijij

)()()1()( qqqq paww

)(...,),2(),1( Qppp

Ex: Banana Associator

Initial weights:

Training sequence:

Learning rule:

0)0(,10 ww

}...1)3(,0)3({},1)2(,1)2({},1)1(,0)1({ 000 pppppp

1),()()1()( qpqaqwqw

n a0p 0w

Shape Smell

Network

Banana ?

First iteration (sight fails):

(no response)

Second iteration (sight works):

(banana)

0)5.01001(

)5.0)1()0()1(()1( 00

hardlim

pwpwhardlima

0100)1()1()0()1( paww

1)5.01011(

)5.0)2()1()2(()2( 00

hardlim

pwpwhardlima

1110)2()2()1()2( paww

Third iteration (sight fails):

(banana)

From now on, the network is capable of responding to bananas that are detected either sight or smell. Even if both detection systems suffer intermittent faults, the network will be correct most of the time.

1)5.01101(

)5.0)3()2()3(()3( 00

hardlim

pwpwhardlima

2111)3()3()2()3( paww

Problems of Hebb Rule

Weights will become arbitrarily large

Synapses cannot grow without bound.

There is no mechanism for weights to decrease

If the inputs or outputs of a Hebb network experience ant noise, every weight will grow (however slowly) until the network responds to any stimulus.

Hebb Rule with Decay

, the decay rate, is a positive constant less than one.

This keeps the weight matrix from growing without bound, which can be found by setting both ai and pj to 1, i.e.,

The maximum weight value is determined by the decay rate .

)()()1()1(

)1()()()1()(

qqqqqT

max)1( ijjiijij wpaww

maxijw

First iteration (sight fails): no response

Second iteration (sight works): banana

Third iteration (sight fails): banana

.1.0,1),()()1()1()( qpqaqwqw

0)5.0)1()0()1(()1( 00 pwpwhardlima

001.0100)0(1.0)1()1()0()1( wpaww

1)5.0)2()1()2(()2( 00 pwpwhardlima

101.0110)1(1.0)2()2()1()2( wpaww

1)5.0)3()2()3(()3( 00 pwpwhardlima

9.111.0111)2(1.0)3()3()2()3( wpaww

0 10 20 300

Hebb Rule Hebb with Decay

0 10 20 300

Prob. of Hebb Rule with Decay

Associations will decay away if stimuli are not occasionally presented.

If ai = 0, then

If = 0.1, this reducesto

The weight decays by10% at each iterationfor which ai = 0(no stimulus)

)1()1()( qwqw ijij

)1(9.0)( qwqw ijij

Instar (Recognition Network)

A neuron that has a vector input and a scalar output is referred to as an instar.

This neuron is capable of pattern recognition.

Instar is similar to perceptron, ADALINE and linear associator.

Instar Operation

Input-output expression:

The instar is active when or

where is the angle between two vectors.

If , the inner product is maximized when the angle is 0.

Assume that all input vectors have the same length (norm).

)()( 1 bhardlimbhardlima T pwWp

bT pw1 bT cos11 pwpw

Vector Recognition

If , then the instar will be only active when = 0.

If , then the instarwill be active for a range of angles.

The larger the value of b, the more patterns there will be that can activate the instar, thus making it the less discriminatory.

Forgetting problem in Hebb rule with decay: it requires stimuli to be repeated or associations would be lost.

pw1bw1

Instar Rule

Hebb rule:

Hebb rule with decay:

Instar rule: a decay term, the forgetting problem, is add that is proportion to

(a rule allow weight decay only when the instar is active (a !=0)

)()()1()( qpqaqwqw jiijij

)()()1()1()( qpqaqwqw jiijij

)1()()()()1()( qwqaqpqaqwqw ijijiijij

)]1()()[()1()(

qqqaqq

qwqpqaqwqw

ijjiijij

Graphical Representation

For the case where the instar is active( ),

For the case where the instaris inactive ( ),

)()1()1(

)]1()([)1()(

)1( qi w

)(qi w0ia

)1()( qq ii ww

Ex: Orange Recognizer

The elements of p will be contained to 1 values.

0p30 w

Sight of orange

Measured shape

Measured texture

Measured weight

Orange?

weight

texture

detectednot orange ,0

visuallydetected orange ,10p

Network

Orange ?

Measure

)( 00 bpwhardlima Wp

b = -2, a value slightly more positive than –||p||2 = -3

Initialization & Training

Initial weights:

The instar rule (=1):

Training sequence:

First iteration:

000)0()0(,3 10 Tw wW

)]1()()[()1()( 111 qqqaqq wpww

)2(,1)2(,

)1(,0)1( 00

response) (no0)2)1()1(()1( 00 Wppwhardlima

Ta 000)]0()1()[1()0()1( 111 wpww

Second Training Iteration

Second iteration:

The network can now recognition the orange by its measurements.

(orange)12

)2)2()2(()2( 00

hardlim

pwhardlima Wp

)]1()2()[2()1()2( 111 wpww a

Third Training Iteration

Third iteration:

(orange)12

)2)3()3(()3( 00

hardlim

pwhardlima Wp

)]2()3()[3()2()3( 111 wpww a

Orange will now be detected if either set of sensors works.

Consider the instar network shown in slide 64, the reaining sequence for this network will consist of following inputs:

{p0(1) = 0, p(1)= }, {p0(2) = 1, p(2)= },

These two sets of inputs are repeatedly presented to the network until the weight matrix w converges.

1. Perform the first four iterations of the instar rule, with learning rate =0.5, Assume that the initial w maytrix is set to all zeros.

2. Display the results of each iteration of the instar rule in graphical form.

Kohonen Rule

Kohonen rule:

Learning occurs when the neuron’s index i is a member of the set X(q).

The Kohonen rule can be made equivalent to the instar rule by defining X(q) as the set of all i such that

The Kohonen rule allows the weights of a neuron to learn an input vector and is therefore suitable for recognition applications.

)()],1()([)1()( 111 qXiqqqq wpww

1)( qai

Ourstar (Recall Network)

The outstar network has a scalar input and a vector output.

It can perform pattern recall by associating a stimulus with a vector response.

Outstar Operation

Input-output expression:

If we would like the outstar network to associate a stimulus (an input of 1) with a particular output vector a*, set W = a*.

If p = 1, a = satlins(Wp) = satlins(a*p) = a* Hence, the pattern is correctly recalled.

The column of a weight matrix represents the pattern to be recalled.

)( psatlinsa W

Outstar Rule

In instar rule, the weight decay term of Hebb rule is proportional to the output of network, ai.

In outstar rule, the weight decay term of Hebb rule is proportional to the input of network, pj.

If = ,

Learning occurs whenever pj is nonzero (instead of ai). When learning occurs, column wj moves toward the output vector. (complimentary to instar rule)

)1()()()()1()( qwqpqpqaqwqw ijjjiijij

)()]1()([)1()(

qpqqqq

qpqwqaqwqw

jijiijij

Ex: Pineapple Recaller

Any set of p0 (with 1 values) will be copied to a.

Network

Measurement?

Measure

Measured shape

Measured texture

Measured weight

Identified Pineapple 3n 3a

Recalled shape

Recalled texture

Recalled weight

)( 00 ppsatlins WWa

weight

texture

shape0p

otherwise ,0seen becan pineapple a if ,1

Initialization

The outstar rule (=1):

Training sequence:

Pineapple measurements:

)()]1()([)1()( qpqqqq jjjj waww

,1)2(,

)2(,1)1(,

)1( 00

1pineapplep

First Training Iteration

First iteration:

response) (no

)1()1()1( 00

satlins

ppsatlins WWa

)1()]0()1([)0()1( 111 pwaww

Second Training Iteration

Second iteration:

The network forms an association between the sight and the measurements.

given) nts(measureme

satlinsa

)2()]1()2([)1()2( 111 pwaww

Third Training Iteration

Third iteration:

Even if the measurement system fail, the network is now able to recall the measurements of the pineapple when it sees it.

recalled) nts(measureme

satlinsa

)3()]2()3([)2()3( 111 pwaww

1 back-propagation. 2 objectives a generalization of the lms algorithm, called backpropagation, can...

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