CS407 Neural Computation · 2004-09-03 · The first layer, termed input layer, just contains the...

CS407 Neural Computation

Lecture 5: The Multi-Layer Perceptron (MLP)

and Backpropagation

Lecturer: A/Prof. M. Bennamoun

What is a perceptron and what is a Multi-Layer Perceptron (MLP)?

What is a perceptron?

... ... Σ

Biasbk

ϕ(.)vk

Inputsignal

Synapticweights

Summingjunction

Activationfunction

Outputyk

bxwv kj

jkjk += ∑

)(vy kkϕ=

)()( ⋅=⋅ signϕDiscrete Perceptron:

shapeS −=⋅)(ϕContinous Perceptron:

Activation Function of a perceptron

Signum Function (sign)

)()( ⋅=⋅ signϕDiscrete Perceptron: shapesv −=)(ϕ

Continous Perceptron:

MLP Architecture The Multi-Layer-Perceptron was first introduced by M. Minsky and S. Papertin 1969

Type: Feedforward

Neuron layers: 1 input layer 1 or more hidden layers 1 output layer

Learning Method: Supervised

Terminology/ConventionsArrows indicate the direction of data flow.

The first layer, termed input layer, just contains the input vector and does not perform any computations.

The second layer, termed hidden layer, receives input from the input layer and sends its output to the output layer.After applying their activation function, the neurons in

the output layer contain the output vector.

Why the MLP?The single-layer perceptron classifiers discussed previously can only deal with linearly separable sets of patterns.

The multilayer networks to be introduced here are the most widespread neural network architecture– Made useful until the 1980s, because of lack

of efficient training algorithms (McClelland and Rumelhart 1986)

– The introduction of the backpropagationtraining algorithm.

Different Non-Linearly Separable Problems http://www.zsolutions.com/light.htm

StructureTypes of

Decision RegionsExclusive-OR

ProblemClasses with

Meshed regionsMost General

Region Shapes

Single-Layer

Two-Layer

Three-Layer

Half PlaneBounded ByHyperplane

Convex OpenOr

Closed Regions

Arbitrary(Complexity

Limited by No.of Nodes)

What is backpropagation Training

and how does it work?

Supervised Error Back-propagation Training– The mechanism of backward error transmission

(delta learning rule) is used to modify the synaptic weights of the internal (hidden) and output layers

• The mapping error can be propagated into hidden layers

– Can implement arbitrary complex/output mappings or decision surfaces to separate pattern classes

• For which, the explicit derivation of mappings and discovery of relationships is almost impossible

– Produce surprising results and generalizations

What is Backpropagation?

Architecture: Backpropagation NetworkThe Backpropagation Net was first introduced by G.E. Hinton, E. Rumelhartand R.J. Williams in 1986

Type: Feedforward

Neuron layers: 1 input layer 1 or more hidden layers 1 output layer

Learning Method: Supervised

Reference: Clara Boyd

Backpropagation PreparationTraining SetA collection of input-output patterns that are used to train the networkTesting SetA collection of input-output patterns that are used to assess network performanceLearning Rate-αA scalar parameter, analogous to step size in numerical integration, used to set the rate of adjustments

Backpropagation training cycle

1/ Feedforward of the input training pattern

2/ Backpropagation of the associated error3/ Adjustement of the weights

Reference Eric Plammer

Backpropagation Neural NetworksArchitectureBP training AlgorithmGeneralizationExamples – Example 1– Example 2

Uses (applications) of BP networksOptions/Variations on BP– Momentum– Sequential vs. batch– Adaptive learning rates

AppendixReferences and suggested reading

ArchitectureBP training AlgorithmGeneralizationExamples – Example 1– Example 2

15Source: Fausett, L., Fundamentals of Neural Networks, Prentice Hall, 1994.

Notation -- p. 292 of FausettNotation -- p. 292 of Fausett

BP NN With Single Hidden Layer

I/P layer

O/P layer

Hidden layer

Reference: Dan St. ClairFausett: Chapter 6

Notationx = input training vectort = Output target vector.δk = portion of error correction weight for wjk that is due

to an error at output unit Yk; also the information about the error at unit Yk that is propagated back to the hidden units that feed into unit Yk

δj = portion of error correction weight for vjk that is due to the backpropagation of error information from the output layer to the hidden unit Zj

α = learning rate.voj = bias on hidden unit jwok = bias on output unit k

Hyberbolictangent

Binary stepActivation Functions

)](1[*)()(

)exp(11)(

' xfxfxf

Should be continuos, differentiable, and monotonically non-decreasing. Plus, its derivative should be easy to compute.

19Fausett, L., pp. 294-296.

Z1 Zj Z3

X1 X2 X3

20Fausett, L., pp. 294-296.

Z1 Zj Z3

X1 X2 X3

21Fausett, L., pp. 294-296.

Z1 Zj Z3

X1 X2 X3

22Fausett, L., pp. 294-296.

Z1 Zj Z3

X1 X2 X3

Let’s examine Training Algorithm Equations

[ ]nxxX ...1=

[ ]pvvV ,01,00 ...=

Z1 Z2 Z3

X1 X2 X3

Vectors & matrices make computation easier.

............

[ ])_(...)_(_

pinzfinzfZXVVinZ

Step 4 computation becomes

[ ]mwwW ,01,00 ...=

............

Step 5 computation becomes

[ ])_(...)_(_

minyfinyfYZWWinY

GeneralisationOnce trained, weights are held contstant, and input patterns are applied in feedforwardmode. - Commonly called “recall mode”.We wish network to “generalize”, i.e. to make sensible choices about input vectors which are not in the training setCommonly we check generalization of a network by dividing known patterns into a training set, used to adjust weights, and a test set, used to evaluate performance of trained network

Generalisation …Generalisation can be improved by– Using a smaller number of hidden units

(network must learn the rule, not just the examples)

– Not overtraining (occasionally check that error on test set is not increasing)

– Ensuring training set includes a good mixture of examples

No good rule for deciding upon good network size (# of layers, # units per layer)Usually use one input/output per class rather than a continuous variable or binary encoding

Example 1

The XOR function could not be solved by a single layer perceptron network

The function is:

X Y F0 0 00 1 11 0 11 1 0

Reference: R. Spillman

XOR Architecture

fv11 Σv01

fv12 Σv02

fw11 Σw01

Initial WeightsRandomly assign small weight values:

f.21 Σ-.3

f-.4 Σ.25

f-.2 Σ-.4

Feedfoward – 1st Pass

f.21 Σ-.3

f-.4 Σ.25

f-.2 Σ-.4

Training Case: (0 0 0)

zin1 = -.3(1) + .21(0) + .15(0) = -.3

Activation function f:

z1 = .43

zin2 = .25(1) -.4(0) + .1(0)

z2 = .56

1yin1 = -.4(1) - .2(.43)

+.3(.56) = -.318

y1 = .42(not 0)

xexf −+

Backpropagate

f.21 Σ-.3

f-.4 Σ.25

f-.2 Σ-.4

δ1 = (t1 – y1)f’(y_in1)=(t1 – y1)f(y_in1)[1- f(y_in1)]

δ1 = (0 – .42).42[1-.42]= -.102

δ_in1 = δ1w11 = -.102(-.2) = .02δ1 = δ_in1f’(z_in1) = .02(.43)(1-.43)

= .005

δ_in2 = δ1w21 = -.102(.3) = -.03δ2 = δ_in2f’(z_in2) = -.03(.56)(1-.56)

= -.007

Calculate the Weights – First Pass

f.21 Σ-.3

f-.4 Σ.25

f-.2 Σ-.4

11 2,1

wjzw jj

102.0 −=∆w

0571.)56)(.102.(2121 −=−==∆ zw δ

0439.)43)(.102.(1111 −=−==∆ zw δ

005.01 −=∆v

007.02 −=∆v

0)0)(005(.1111 ===∆ xv δ

0)0)(007.(1212 =−==∆ xv δ

0)0)(005(.2121 ===∆ xv δ

0)0)(007.(2222 =−==∆ xv δ

Update the Weights – First Pass

f.21 Σ-.305

f-.4 Σ.243

f-.044Σ-.502

Final ResultAfter about 500 iterations:

f1 Σ-1.5

f1 Σ-.5

f-2 Σ-.5

Example 2

[ ]08.06.0=X

[ ]10 −=w[ ]1000 −=v

130221012

Z1 Z2 Z3

X1 X2 X3

α = 0.3

Desired output for X input

)1(1)( xe

xf −+=

Reference: Vamsi Pegatraju and Aparna Patsa

Primary Values: Inputs to Epoch - IX=[0.6 0.8 0];W=[-1 1 2]’;W0=[-1];V= 2 1 0

1 2 2 0 3 1

V0=[ 0 0 -1];Target t=0.9;α = 0.3;

Epoch – IStep 4: Z_in= V0+XV = [ 2 2.2 0.6];

Z=f([Z_in])=[ 0.8808 0.9002 0.646];Step 5: Y_in = W0+ZW = [0.3114];

Y=f([Z_in])=0.5772;Sum of Squares Error obtained originally:(0.9 – 0.5772)2 = 0.1042

Step 6: Error = tk – Yk = 0.9 – 0.5772Now we have only one output and hence the value of k=1.δ1= (t1 – y1 )f’(Y_in1)We know f’(x) for sigmoid = f(x)(1-f(x))

⇒ δ1 = (0.9 −0.5772)(0.5772)(1−0.5772)= 0.0788

For intermediate weights we have (j=1,2,3)∆Wj,k=α δκΖj = α δ1Ζj

⇒ ∆W1=(0.3)(0.0788)[0.8808 0.9002 0.646]’=[0.0208 0.0213 0.0153]’;

Bias ∆W0,1=α δ1= (0.3)(0.0788)=0.0236;

Step 7: Backpropagation to the first hidden layerFor Zj (j=1,2,3), we haveδ_inj = ∑k=1..m δκWj,k= δ1Wj,1

⇒ δ_in1=-0.0788;δ_in2=0.0788;δ_in3=0.1576;δj= δ_injf’(Z_inj)

=> δ1=-0.0083; δ2=0.0071; δ3=0.0361;

∆Vi,j = αδjXi

⇒ ∆V1 = [-0.0015 -0.0020 0]’;⇒ ∆V2 = [0.0013 0.0017 0]’;⇒ ∆V3 = [0.0065 0.0087 0]’;

∆V0=α[δ1 δ2 δ3] = [ -0.0025 0.0021 0.0108];

X=[0.6 0.8 0]

Step 8: Updating of W1, V1, W0, V0

Wnew= Wold+∆W1=[-0.9792 1.0213 2.0153]’;Vnew= Vold+∆V1

=[1.9985 1.0013 0.065;0.998 2.0017 2.0087;0 3 1];

W0new = -0.9764;V0new = [-0.0025 0.0021 -0.9892];Completion of the first epoch.

Primary Values: Inputs to Epoch - 2X=[0.6 0.8 0];W=[-0.9792 1.0213 2.0153]’;W0=[-0.9792];V=[1.9985 1.0013 0.065; 0.998 2.0017 2.0087;

0 3 1]; V0=[ -0.0025 0.0021 -0.9892];Target t=0.9;α = 0.3;

Epoch – 2Step 4:Z_in=V0+XV=[1.995 2.2042 0.6217];Z=f([Z_in])=[ 0.8803 0.9006 0.6506];

Step 5: Y_in = W0+ZW = [0.3925];Y=f([Z_in])=0.5969;

Sum of Squares Error obtained from first epoch: (0.9 – 0.5969)2 = 0.0918

Step 6: Error = tk – Yk = 0.9 – 0.5969Now again, as we have only one output, the value of k=1.δ1= (t1 – y1 )f’(Y_in1)

=>δ1 = (0.9 −0.5969)(0.5969)(1−0.5969)= 0.0729

For intermediate weights we have (j=1,2,3)∆Wj,k=α δκΖj = α δ1Ζj

⇒ ∆W1=(0.3)*(0.0729)*[0.8803 0.9006 0.6506]’

=[0.0173 0.0197 0.0142]’;Bias ∆W0,1=α δ1= 0.0219;

Step 7: Backpropagation to the first hidden layerFor Zj (j=1,2,3), we haveδ_inj = ∑k=1..m δκWj,k= δ1Wj,1

⇒ δ_in1=-0.0714;δ_in2=0.0745;δ_in3=0.1469;δj= δ_injf’(Z_inj)

=> δ1=-0.0075; δ2=0.0067; δ3=0.0334;

∆Vi,j = αδjXi

⇒ ∆V1 = [-0.0013 -0.0018 0]’;⇒ ∆V2 = [0.0012 0.0016 0]’;⇒ ∆V3 = [0.006 0.008 0]’;

∆V0=α[δ1 δ2 δ3] = [ -0.0022 0.002 0.01];

Step 8: Updating of W1, V1, W0, V0

Wnew= Wold+∆W1=[-0.9599 1.041 2.0295]’;Vnew= Vold+∆V1

=[1.9972 1.0025 0.0125; 0.9962 2.0033 2.0167; 0 3 1];

W0new = -0.9545;V0new = [-0.0047 0.0041 -0.9792];

Completion of the second epoch.

Z_in=V0+XV=[1.9906 2.2082 0.6417];=>Z=f([Z_in])=[ 0.8798 0.9010 0.6551];

Step 5: Y_in = W0+ZW = [0.4684];=> Y=f([Z_in])=0.6150;

Sum of Squares Error at the end of the second epoch: (0.9 – 0.615)2 = 0.0812.From the last two values of Sum of Squares Error, we see that the value is gradually decreasing as the weights are getting updated.

Functional ApproximationMulti-Layer Perceptrons can approximate any continuous function by a two-layer network with squashing activation functions.If activation functions can vary with the function, can show that a n-input, m-output function requires at most 2n+1 hidden units.See Fausett: 6.3.2 for more details.

Function ApproximatorsExample: a function h(x) approximated by H(w,x)

ApplicationsWe look at a number of applications for

backpropagation MLP’s.In each case we’ll examine–Problem to be solved–Architecture Used–Results

Reference: J.Hertz, A. Krogh, R.G. Palmer, “Introduction to the Theory of Neural Computation”, Addison Wesley, 1991

NETtalk - SpecificationsProblem is to convert written text to speech.Conventionally, this is done by hand-coded linguistic rules, such as the DECtalk system. NETtalk uses a neural network to achieve similar resultsInput is written textOutput is choice of phoneme for speech synthesiser

NETtalk - architecture

e c ah t oT n7 letter sliding window, generatingphoneme for centre character.Input units use 1 of 29 code.=> 203 input units (=29x7)

80 hidden units, fully interconnected

26 output units, 1 of 26 code representing most likely phoneme

NETtalk - Results

1024 Training SetAfter 10 epochs - intelligible speechAfter 50 epochs - 95% correct on training set

- 78% correct on test setNote that this network must generalise - many input combinations are not in training setResults not as good as DECtalk, but significantly less effort to code up.

Sonar Classifier

Task - distinguish between rock and metal cylinder from sonar return of bottom of bayConvert time-varying input signal to frequency domain to reduce input dimension.(This is a linear transform and could be done with a fixed weight neural network.)Used a 60-x-2 network with x from 0 to 24Training took about 200 epochs. 60-2 classified about 80% of training set; 60-12-2 classified 100% training, 85% test set

ALVINNDrives 70 mph on a public highway

30x32 pixelsas inputs

30 outputsfor steering

30x32 weightsinto one out offour hiddenunit

4 hiddenunits

Navigation of a Car

Task is to control a car on a winding roadInputs are a 30x32 pixel image from a video camera on roof, 8x32 image from a range finder => 1216 inputs29 hidden units45 output units arranged in a line, 1-of-45 code representing hard-left..straight-ahead..hard-right

Navigation of Car - Results

Training set of 1200 simulated road imagesTrained for 40 epochsCould drive at 5 km/hr on road, limited by calculation speed of feed-forward network.Twice as fast as best non-net solution

Backgammon

Trained on 3000 example board scenarios of (position, dice, move) rated from -100 (very bad) to +100 (very good) from human expert.Some important information such as “pip-count” and “degree-of-trapping” was included as input.Some “noise” added to input set (scenarios with random score)Handcrafted examples added to training set to correct obvious errors

Backgammon results

459 inputs, 2 hidden layers, each 24 units, plus 1 output for score (All possible moves evaluated)Won 59% against a conventional backgammon program (41% without extra info, 45% without noise in training set)Won computer olympiad, 1989, but lost to human expert (Not surprising since trained by human scored examples)

Encoder / Image Compression

Wish to encode a number of input patterns in an efficient number of bits for storage or transmissionWe can use an autoassociative network, i.e. an M-N-M network, where we have M inputs, and N<M hidden units, M outputs, trained with target outputs same as inputsHidden units need to encode inputs in fewer signals in the hidden layers.Outputs from hidden layer are encoded signal

Encoders

We can store/transmit hidden values using first half of network; decode using second half.We may need to truncate hidden unit values to fixed precision, which must be considered during training.Cottrell et al. tried 8x8 blocks (8 bits each) of images, encoded in 16 units, giving results similar to conventional approaches.Works best with similar images

Neural network for OCRfeedforward networktrained using Back-propagation

Output Layer

Input Layer

Hidden Layer

Pattern Recognition

Post-code (or ZIP code) recognition is a good example - hand-written characters need to be classified.One interesting network used 16x16 pixel map input of handwritten digits already found and scaled by another system. 3 hidden layers plus 1-of-10 output layer.First two hidden layers were feature detectors.

ZIP code classifier

First layer had same feature detector connected to 5x5 blocks of input, at 2 pixel intervals => 8x8 array of same detector, each with the same weights but connected to different parts of input.Twelve such feature detector arrays.Same for second hidden layer, but 4x4 arrays connected to 5x5 blocks of first hidden layer; with 12 different features.Conventional 30 unit 3rd hidden layer

ZIP Code Classifier - Results

Note 8x8 and 4x4 arrays of feature detectors use the same weights => many fewer weights to train.Trained on 7300 digits, tested on 2000Error rates: 1% on training, 5% on test setIf cases with no clear winner rejected (i.e. largest output not much greater than second largest output), then, with 12% rejection, error rate on test set reduced to 1%.Performance improved further by removing more weights: “optimal brain damage”.

Heuristics for making BP BetterTraining with BP is more an art than science– result of own experience

Normalizing the inputs– preprocessed so that its mean value is

closer to zero (see “prestd” function in matlab).

– input variables should be uncorrelated• by “Principal Component Analysis” (PCA). See

“prepca” and “trapca” functions in Matlab.

Sequential vs. Batch update

“Sequential” learning means that a given input pattern is forward propagated, the error is determined and back-propagated, and the weights are updated. Then the same procedure is repeated for the next pattern.“Batch” learning means that the weights are updated only after the entire set of training patterns has been presented to the network. In other words, all patterns are forward propagated, and the error is determined and back-propagated, but the weights are only updated when all patterns have been processed. Thus, the weight update is only performed every epoch.

If P = # patterns in one epoch

∆=∆P

Sequential vs. Batch updatei.e.in some cases, it is advantageous to accumulate the weight correction terms for several patterns (or even an entire epoch if there are not too many patterns) and make a single weight adjustment (equal to the average of the weight correction terms) for each weight rather than updating the weights after each pattern is presented. This procedure has a “smoothing effect”(because of the use of the average) on the correction terms. In some cases, this smoothing may increase the chances of convergence to a local minimum.

Initial weightsInitial weights – will influence whether the net reaches

a global (or only a local minimum) of the error and if so, how quickly it converges.– The values for the initial weights must not be too large otherwise,

the initial input signals to each hidden or output unit will be likely to fall in the region where the derivative of the sigmoid function has a very small value (f’(net)~0) : so called saturation region.

– On the other hand, if the initial weights are too small, the net input to a hidden or output unit will be close to zero, which also causes extremely slow learning.

– Best to set the initial weights (and biases) to random numbers between –0.5 and 0.5 (or between –1 and 1 or some other suitable interval).

– The values may be +ve or –ve because the final weights after training may be of either sign also.

Memorization vs. generalizationHow long to train the net: Since the usual motivation for applying a backprop net is to achieve a balance between memorization and generalization, it is not necessarily advantageous to continue training until the error actually reaches a minimum.

– Use 2 disjoint sets of data during training: 1/ a set of training patterns and 2/ a set of training- testingpatterns (or validation set).

– Weight adjustment are based on the training patterns; however, at intervals during training, the error is computed using the validation patterns.

– As long as the error for the validation decreases, training continues.

– When the error begins to increase, the net is starting to memorize the training patterns too specifically (starts to loose its ability to generalize). At this point, training is terminated.

Early stoppingError

Training time

With training set (which changes wij)

With validation set (which does not change wij)

Stop Here !

L. Studer, IPHE-UNIL

Backpropagation with momentumBackpropagation with momentum: the weight change is in a direction that is a combination of 1/ the current gradient and 2/ the previous gradient.

Momentum can be added so weights tend to change more quickly if changing in the same direction for several training cycles:-∆ wij (t+1) = α δ xi + µ . ∆ wij (t)µ is called the “momentum factor” and ranges from 0 < µ < 1. – When subsequent changes are in the same direction increase

the rate (accelerated descent)– When subsequent changes are in opposite directions decrease

the rate (stabilizes)

Backpropagation with momentum…Weight update equation Momentum

)1( −tw

)(tw z)( αδ+tw

)1( +tw

Source: Fausett, L., Fundamentals of Neural Networks, Prentice Hall, 1994, pg. 305.

Adaptive learning

BP training algorithm –Adaptive Learning Rate

Adaptive Learning rate…Adaptive Parameters: Vary the learning rate during training, accelerating learning slowly if all is well ( error, E, decreasing) , but reducingit quickly if things go unstable (E increasing).

For example:

Typically, a = 0.1, b = 0.5

<∆+=+

otherwise (t)0 E if (t) . b)-(1

epochs fewlast for 0 E if a (t) 1)(t

Matlab BP NN ArchitectureA neuron with a single R-element input vector is shown below. Here the individual element inputs

•are multiplied by weights

•and the weighted values are fed to the summing junction. Their sum is simply Wp, the dot product of the (single row) matrix W and the vector p.

The neuron has a bias b, which is summed with the weighted inputs to form the net input n. This sum, n, is the argument of the transfer function f.

This expression can, of course, be written in MATLAB code as:•n = W*p + b

However, the user will seldom be writing code at this low level, for such code is already built into functions to define and simulate entire networks.

Matlab BP NN Architecture

Learning RuleSimilar to Delta Rule.Our goal is to minimize the error, E, which is the difference between targets, tm , and our outputs, yk

m , using a least squares error measure: E = 1/2 Σk (tk - yk)2

To find out how to change wjk and vij to reduce E, we need to find

ijjk vEand

∂∂

Fausett, section 6.3, p324

Delta Rule Derivation Hidden-to-Output

−=−= ∑∑

2 )y(t21

wwEhence][5.0

∂∂

kkk ytE

[ ] [ ]

−= ∑ 2

inKk yfy∂

∂∂

∂∂∂

and)( where ∑==j

jKjinKinKk wzyyfy

JKin KKJK

).z(y')fy(twE

−−=∂∂

∂∂

−−=∂

∂−−=

)().(y')fy(t)()y(twE

Kin KKKKJK∂

Notice the difference between the subscripts k (which corresponds to any node between hidden and output layers) and K (which represents aparticular node K of interest)

Delta Rule Derivation Hidden-to-Output

)(y')f(t :define toconvenient isIt inKK KK y−=δ

jkjinkkk zzyfyt δαα∂∂α =−=−=∆ )('][w

Ew Thus,jk

jk zδα=∆ jkw summary,In )(y')f(twith inKK KK y−=δ

Delta Rule Derivation: Input to Hidden

inkink

yyfytvyyt

v ∂∂

−−=∂∂

−−= ∑∑ )('][][E

IJ∂∂

])[('vE

IJIinJJk

kk xzfw

vy ∑∑∑ −=

∂∂

−=∂∂

−= δδδ∂∂

−=−= ∑∑

2 )y(t21

vEhence][5.0

kkk ∂

∂∂∂

and)( where ∑==j

jKjinKinKk wzyyfy

)(z'f :define toconvenient isIt inJ∑=k

JkkJ wδδ

Notice the difference between the subscripts j and J and i and I

jkkiinjij xwxzfv αδδα∂∂α ==−=∆ ∑)('

Delta Rule Derivation: Input to Hidden

)(z'f :where inJ∑=k

JkkJ wδδijij xv αδ=∆ summary In

CS407 Neural Computation · 2004-09-03 · The first layer, termed input layer, just contains the...

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