continuous hopfield

7/25/2019 Continuous Hopfield

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Optimization with Neural

Networks

Presented by:

Nasim Zeynolabedinishoale HashemiZahra Rashti

Instructor:

Dr. S. Bagheri

Sharif University of Technoloy

Ordibehesht !"#$


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Introduction

Optimization Problem:

% & Problem with a cost function that is to be

minimized or ma'imized

e: TSP( )napsack( *raph Partitionin( *raph+isection( *raph ,olorin( etc-

% .any solutions to solve these problems such as:

/inear optimization( Simulated &nnealin( .ont

,arlo( &N0Neural Networks-


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Applications

&pplications in many fields like:

1outin in computer networks

2/SI circuit desin Plannin in operational and loistic systems Power distribution systems 3ireless and satellite communication systems


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Optimization roblems !ypes

&n optimization problem consists of two parts:,ost

functionand,onstraints

% ,onstrained

4 The constraints are built in the cost function( sominimizin the cost function also satisfies the constraints

% Unconstraint

4 There is no constraint for the problem5

% ,ombinatorial

4 3e separate the constraints and the cost function( minimize

each of them and then add them toether


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"hy Neural Net#or$s%

% 0rawbacks of conventional computin systems:

4 Perform poorly on comple' problems

4 /ack the computational power

4 0ont utilize the inherent parallelism of problems

% &dvantaes of artificial neural networks:

4 Perform well even on comple' problems

4 2ery fast computational cycles if implemented in hardware

4 ,an take the advantae of inherent parallelism of problems


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Some &''orts to Sol(e Optimization roblems

% .any &NN alorithms with feedforward and recurrent

architectures have been used to solve different

optimization problems

% 3eve selected:

4 6opfield NN

4 Self Oranizin .ap NN

4 1ecurrent NN to solve TSP as the most common benchmark for

optimization alorithms-


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,ontinuous 6opfield

Neuron function is continuous (Sigmoid function)

System behavior is described by

a differential equation :

i

n

j

jijii Vw

U

dt

dU

++=

=!

iUii eUfV

+

==

!

!78


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&n electronic implementation


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+asic idea

If : decision variables

Suppose is our obective function !

"onstraints can be e#pressed as nonnegative

penalty functions that only $hen

represent a feasible solution

%y combining the penalty functions $ith & ' the

original constrained problem may be reformulatedas unconstrained problem in $hich the goal is to

minimie the quantity :

nXXX (---(( 9!

79! (---((8 nXXXF

nXXX (---(( 9!

7(---((8 9! ni XXXC

:7(---((8 9! =ni XXXC

=

+=

m

knkn XXXCXXXFF

!9!9! 7(---((87(---((8


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+asic idea 8cont-7

* is a sufficiently large scaling factor for the

penalty terms !

+inimiing yields a minimal ' feasible solution

to the original problem &urthermore if can be $ritten in the form of

energy function ' there is a corresponding neural

net$or, $hose equilibria represent solution to the

problem

F

F


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Simplification of enery function

is a -yapunov function so long as the the

function is sigmoidal!

.e modify slightly' but in such a $ay that it

remains sigmoidal /78 iii ufV = 78 iii ufV =


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Simplification of enery function 8cont-7

0he inverse function can obviously be

If $e use this in the middle term of energy function :

if $e let become very large' this term $ill become

negligible! 0he function is still a sigmoid' so L(v) is

still a -yapunov function' but in this situation (,no$n asthe high-gain limit) the middle term can be ignored!

787;!8 ! iii Vfu

=


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Travelin Salesman Problem

"hec,ing out all possible routes :

routes

N / 1* / routes

route in a second / seconds

/ 123*4342456 years

In many industrial problems : N 1*

7 continuous 8opfield net$or, can be

constructed to quic,ly provide a good solution to

the 0S9


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0he 8opfield net$or, approach to the 0S9 involvesarranging the net$or, neurons in such a $ay that theyrepresent the entries in the table

for an Ncity problem $e $ould require neurons

N of them $ill be turned ;N $ith the remainder turned;&&


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Ob


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Settin up the function to be minimized

% : state 8 or !7 of the neuron in position 8i(a7 in

the table

% : distance between city i and cityj

% Total lenth of tour :

iaV

ijd


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% If we take into account both constraints and ob


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Finding the network weights and input currents

% 3e should select weihts and currents so that two

followin e>uations become e>ual

% ?irst we make output voltaes double subscripts :


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% Note first that the multiply second=order termsand the multiply first order terms

-

% ?irst order terms should be e>ual :

jbiaw (jbiaVV iai

iaV


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% we will need to treat the four sets of second=order termsin separately

the second=order C terms are iven by

@ = ,


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% If then will add to

lyapunov function but what we want is

)ronecker delta :

So we should have

In the same way :

AwA bjia =(

elseif! jiij ==


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% D term contributes an amount to the /yapunovfunction only when or when

so :

+rinin toether all four components of the weihts(

we have( finally:


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Applying the method in practice

% suitable values for the parametersA(B(C,D and must

be determined

% Tank and 6opfield usedA@B@D@9A ( ,@! and

@A% Tank and 6opfield applied it to random !=city maps

and found that( overall( in about AB of cases( the

method found the optimum route from amon the

!#!$$ distinct paths


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% The size of each blacks>uare indicates the valueof the output of thecorrespondin neuron


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Self Oranizin .ap NN

% In !CDA Teuvo )ohonen introduced new type of neural

network that uses competitive( unsupervised learnin


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Self Oranizin .ap NNSummary o' the algorithm

1.Initialization) ,hoose random values for the initial weiht vectors wj87-

2.Sampling)Select an input e'amplex from the trainin set for use as an input-

3.Identify winning neuron)?ind the neuron whose weiht vector is closest to the inputx.

4.Updating)&dual to zero-5.Termination:,ontinue by returnin to step 9 until there no further chanes in the

feature map-

racticalities

!-The learnin rate parameter should bein with a value close to unity and decreaseradually as learnin proceeds-

9- The neihborhoods should also decrease in size as learnin proceeds-


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Self Oranizin .ap NN

% One dimensional

neihborhood of )ohonen

SO.

% ,lassical two dimensional

neihborhood

% F'tended two dimensionalneihborhood of )ohonenSO.


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Self Oranizin .ap NN% Self=oranization of a network with two dimensional neihborhood-

Sel'*organization o' a net#or$ #ith one dimensional neighborhood.


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TSP Solvin

% The Flastic Net &pproach4 0urbin and 3illshaw first proposed the elastic net method

in !C#D as a means of solvin the TSP-

% SO. &pproach4 Fven before 0urbin and 3illshaws work on the elastic net

method was published( ?ort had been workin on the idea

of usin a self oranizin process to solve the TSP-


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The Flastic Net &pproach


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SO. &pproach


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.odifications of SO.

% The work of &neniol et al- is based on the distinctive feature

that units in the rin are dynamically created and deleted-

% +urke and 0amany use the GconscienceG mechanism to solve

the problem related to the mappin of multiple cities to the

same unit in the rin-

% .atsuyama adds a new term( previously introduced in 0urbin

and 3illshaw to the weiht update e>uations-


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Flastic Net vs- SO.

% The difference between the two approaches however is that

?orts alorithm incorporates stochasticities into the weiht

adaptations whereas the elastic net method is completely

deterministic in nature-

% There is also no enery minimization involved with the

method of ?ort-

% ?ortHs results were not as ood as those obtained usin the

elastic net method-


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Flastic Net vs- SO.8cont-7

FN: Flastic

Net*N *uilty Net

6T 6opfield=Tank


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&dvantaes of a self oranizin approach

% The reater bioloical resemblance of the SO?.

% the reduced number of neurons and synapses needed toperform optimization tasks

% The )ohonen Self=Oranizin .ap is substantially superior to

the ,ontinuous 6opfield Net-


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6opfield &pproach 2s- SO. &pproach

+est route for !A cities 8usin # runs7 is 9!9D km by

,ontinuous 6opfield and !"!! km by )ohonen Self=Oranizin

.ap- The )ohonen path seems optimal( but this has not been

proven-


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Recurrent Neural Net#or$s

% & recurrent neural network 81NN7 is an &rtificial

Neural Network( which has e'ternal inputs in the form

of a vector ( a feedforward function J8-7 8any

feedforward network includin multi=layer perceptronis appropriate7( outputs in the form of vector K and a

feedbackpath( which copies the outputs to inputs-

% The network behavior is based on its history and so

we must think of pattern presentation as it happens intime-


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Recurrent Neural Net#or$s

4Simple 1ecurrent Networks41ecurrent .ultilayer Perceptron

and

4Simultaneous 1ecurrent Networks

F'ternal

Inputs

6idden /ayer8s7 Output /ayer Output

+ank of

0elays


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Simultaneous Recurrent Neural Net#or$s

Simultaneous 1ecurrent Neural Network 8S1N7 is

a feedforward network with simultaneous feedback

from outputs of the network to its inputswithout

any time delay- ?ormal description of S1N:

Feedforwa

rd

+apping

,.-W

Outputs

/eedbac$

ath

Inputs!

7(8 XF!=

7((8

78

7!8

78

X"f"

""!

nn=

=

+


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Simultaneous Recurrent Neural Net#or$s

?ollows a tra


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SRN !raining

The standard procedure to train a recurrent

network is to define an error measure( which is a

function of network outputs and modify the

weihts usin a derivative of the error with respectto the weihts themselves- The eneric weiht

update e>uation is iven by:

ij

o#d

ij

new

ij w

$ww

=


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!raining o' SRN

The derivative values can be computed usin a number of

techni>ues:

+ackpropaation Throuh Time 8+TT7 which re>uires the

knowlede of desired outputs throuhout the trauarantee of yieldin e'act results

in e>uilibrium

Truncation did not provide satisfactory results and needs

to be further tested

1ecurrent +ackpropaation re>uires only knowlede of

desired outputs at the end of tra


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Sol(ing !S #ith SRN

Network topoloy for travelin salesman problem:

6idden

/ayers

,ost

.atri'

Output

&rray

N N

nodes

N N

nodes

N N

nodes

Input /ayer 6idden /ayer8s7 Output /ayer

Path

Specification


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roblem 0onstraints and &rror /unctions

These constraints enforce the row and column

sums to be e>ual to a value of G!-G( to force the

neuron outputs to limitin values of G-G and

G!-G( to eliminate loops in the solution path( andto encourae minimum distance solutions to be

identified


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roblem 0onstraints and &rror /unctions

Frror ?unctions:


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"eight Ad1ustment

The formula used for weiht ad


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"eight Ad1ustment


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"eight arameters

The stability of the network durin trainin reatly

depends on these constraint weiht parameters- 2ery lare

values of these parameters will force all the weihts of the

network to become neative( which tend to make the

network unstable and all the outputs of the network to

convere to -A-

It was also determined throuh e'ploratory

e'perimentation with the alorithm that these parametervalues needed to be chaned durin trainin sub


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!ra(eling Salesman roblem


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&rror 0omputation 'or !S

,onstraints used for trainin TSP

&symmetry of the path traveled

,olumn inhibition 1ow inhibition

,ost of the path traveled

2alues of the solution matri'

Source

0ities

Destination 0ities

Output +atri2


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Simulation)* Initialization 'or !S

Cities 3alues Asymmetry Row/Column Output Value Cost

Initial *!**5 *!**


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Simulation)* !raining

Frror function vs- Simulation Time for TSP


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Simulation)* Results

,onverence criteria of network is checked after every ! rela'ations

,riteria: CAB of active nodes have value reater than -C

Cities NormalizedDistance

between Cities

ComputationalTime inmin/100

Relaxations

Aera!e Numbero" Relaxations "or

#olution

TotalComputational

Time

6* *!=> *!21 23** =!=2 min!

** =3!** min!

=** *!=< 1!=* 6=** 216!6* min!

1** *!=5 3!5= <


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Normalized Distance s$ %roblem #ize

*

*!2

*!=

*!1

*!6

*!

* 6*


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Plot of Number of 1ela'ations re>uired for a solution and values of

,onstraint 3eiht Parameters c and rafter " 1ela'ations vs- Problem Size

%roblem #ize s Number o" Relaxations and

%roblem #ize s Constraint &ei!'t %arameter !cor !

r

*

2***

=***

1***

6***

***

3***

6*


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0onclusions

% The S1N with the 1+P was able to find ood >uality solutions( in the rane of -9A=-"A( for lare=scale 8$ to A city7

Travelin Salesman Problem% Solutions were obtained within acceptable computation efforts

% The simulator developed does not re>uire weihts to bepredetermined before simulation as is the case with the 6N and

its derivatives% The initial and incremental values of constraint weiht

parameters

play very important role in the trainin of the network% ,omputational effort and memory re>uirement increased

proportional to the s>uare of the problem size% The number of rela'ations re>uired increased with the increase

in the problem size


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0onclusions ,continued

Simultaneous 1ecurrent Neural Network with1ecurrent +ackpropaation trainin alorithmscaled #ell

'or large*scale static optimization problemslike the Travelin

Salesman Problem within acceptable computation effort bounds-


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4uestions %

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