On making neural network based learning systems robust

IETE Journal of Research

Vol 44, Nos 4&5, July-October 1998, pp 219-225.

On Making Neural Network Based Learning Systems Robust

ASHISH GHOSH

M achine Intelligence U nit, Indian Statistical Institute, 203 B T Road, C alcu tta 7 0 0 035, India.

ash@ isical.ac.in

AND

HIDEO TANAKA

Department of Industrial Engineering, College of Engineering, Osaka Prefecture University, 1-1 Gakuen-clio,Sakai, Osaka 593, Japan.

tanaka@ie.osakafu-u.ac.jp

A method for making nerual network based learning systems robust with respect to component failure (damaging of nodes/links) is suggested in the present investigation. The method allows some of the components to fail at various instants of the entire learning process.

The change in error value caused by this damage will be adjusted while the other components learn their parameters during the rest part of learning. The damaging/component failure process has been modeled as a Poisson process. The instants or moments of damaging are chosen by statistical sampling. The components to be damaged are determined randomly. As an illustration, the model is implemented on the back-propagation learning algorithm.

Indexing terms: Neural networks, Robustness, Learning algorithms

A

n eural netw ork (N N ) [l' 51 based system consists o f a larg e n u m b er o f neurons w ith m assive connectivity a m o n g them . L ocal connectivity am ong th e neurons/nodes (c o m p u tin g elem ents) being very high and th e storage o f in fo rm a tio n being distributed, th e approach is claim ed to b e h ig h ly ro b u st and can b e applied even w hen in fo rm a tio n is ill-defined and/or defective/partial o r noisy.

I f so m e o f th e com ponents fail to w o rk com pletely or p artially , the o ther com ponents ad ju st them selves (during ite rativ e learn in g ) in such a m a n n er th a t th e outp u t is not d e te rio ra te d m uch. T his featu re is exploited here to p ro v id e a m o d el fo r designing ro b u st N N based learning sy stem s. A s a N N based system co n sists o f a large n um ber o f co m p o n en ts, the possibility o f som e o f its com ponents (n o d e s an d /o r links) to fail to w o rk is very high. H ere lies th e n ec essity o f designing ro b u st (under com plete or p artial failu re o f com ponents) learning system s. In this c o n te x t w e m ention th at several w orks t6' 9* have been d o n e to d esig n o p tim um N N architectures by dam aging so m e o f th e com ponents.

In th e p re se n t w ork an attem p t is m ade to provide a m o d e l fo r d esig n in g robust neural n etw ork based learning system s. T h is is d o n e by d am ag in g (com pletely) som e o f th e co m p o n e n ts o f a N N b ased system during the process o f le arn in g its param eters (w eights and biases) and stu d y in g .the ch an g e in its p erform ance. S ince (som e of) th e co m p o n e n ts are dam aged a t different tim e instants o f

th e entire learning process, th e error values at the output nodes w ill be different than if those com ponents would n o t fail; and thus th e o th e r com ponents will adjust their param eters so as to com pensate for this dam age. T hus the perform ance will not be deteriorated much due to this d am ag e and the system w ill be robust. The com ponent failure process o f N N has already been shown to follow Poisson distribution u n d er certain assum ptions [1°1. U n d er this m odel, the com ponents (nodes and/or links) to be dam aged are chosen random ly. T he tim e instants {i.e., w hen a dam age occurs) are determ ined b y d raw in g ran d o m sam ples from the appropriate prob ab ility distribution I,I1 2 1.

T hough the proposed m odel is valid, in gen eral, for any N N based learning system , the problem o f m ulti-lay er perceptron based classification using back-propagation learning is considered here fo r a dem onstration o f th e validity o f the m odel. It is im plem ented on IR IS d a t a 113'.

To dem onstrate the utility o f th e proposed m odel, a few links are dam aged co m pletely d uring the learning p hase (this has the underlying assum ption that th e tra in in g / learning phase takes a large, am o u n t o f tim e co m p ared to testing phase) o f the classificatory system . In th e te stin g phase, the perform ance" o f such a system is ev aluated by m easuring the p ercen tag e o f co rrect classification. A netw ork with the sam e config u ratio n (as used in th e previous experim ent) is th e n allow ed to learn w ith th e sam e s'et o f training sa m p le s (w ithout any c o m p o n e n t dam age). T he sam e set o f links w hich w ere d a m a g e d

during the learning p hase o f the previous e x p e rim en t are then dam aged (b efo re testing) and th e classification accuracy is m easured. A com parative study o f th e perfect classification rates for these tw o experim ents sh o w s that the proposed m odel provides m ore robust (b etter accuracy even with d am ag in g o f com ponents) perform ance.

MODELING AND SAMPLING OF FAILURES IN A NEURAL NETWORK

Modeling of failures

L et us c o n sid er a neural network system with N com ponents, w here a com ponent could be a node (processor) o r a link. (In practical case one can m odel the nodes and links separately also [l01). During the operation o f the netw ork som e o f its com ponents may fail. We m ake th e follow ing assum ptions about the failure process:

(i) T he system has N identical com ponents at th e tim e instant t = 0 (when the operation o f the netw ork starts).

(it) If a com ponent fails, it fails for ever (no re p a ir or replacem ent).

(Hi) Failure o f com ponents occurs at an average ra te o f fi per unit tim e.

(/v) The probability o f an event occurring betw een tim e t and t + h depends only on the length o f h, i.e., the probability d o es not depend on either the n u m b er o f events that has occurred up to tim e t o r the specific value o f t, i.e., the probability density function has stationary increm ents.

(v) The probability o f a failure during a very sm all interval o f tim e h is positive but less than one (1), i.e., not certain.

(vi) A t m ost one failure can occur during a very sm all interval o f tim e h.

L et p n(t) b e th e probability that the system h as n com ponents active at tim e instant t, i.e., N - n failures during tim e interval [0, f]. It can be shown th at u n d er th e assum ption s (i)-(vi), p n(t) is given by the form ula:

,N -n

Pn(') =

e ^ ‘(jl t)

(N - n)! n = 1 , 2 , . . . , AT

(

¹

)

a n d

Po ( 0 = 1 N

£./»«( o-

n = I

T h u s w e se e th a t p„(t) is a truncated Poisson distribution w ith m ean fi t.

I f f i t ) is the p ro b ab ility density function ( p d f ) o f the inter-failure tim e (i.e., tim e interval betw een tw o

successive failures), then it can be show n that for the earlier. Poisson failure p ro cess,/ ( / ) is given by

f ( t ) = n e -v < t > 0

= 0 t < 0. (3)

Thus, w hen the failure process is gov ern ed by a P o isso n / distribution, the inter-failure tim e is described by ah exponential distribution (3) with expected value (m ean)

E ( 0 = [ ~ t f i t ) d t

•'/i

(4)

Sampling

In .order to sim ulate the failure process, one needs to draw random sam ples from the exponential distribution (3). B efore describing the exact alg o rith m , let us first, consider the general strategy fo r sam p lin g from any distribution.

L e t f ( x ) b e th e p d f o f the random d ev iate x, and F(x) be the cum ulative density function (cdf) o f x , i.e.,

F(X)= r m d t .

(5)

It can be easily show n that th e ran d o m variable y - F ix ) is uniform ly distributed over 10, 1], regardless o f the distribution o f x. H ence, if R is a ran d o m n u m b e r draw n from uniform [0, 1], then x = F ~ liR ) is a random sam ple from the p d f f i x ) . T herefore, sam p lin g from any distribution can be d o n e using the follo w in g sim ple method having tw o steps.

S tep 1: G enerate a random n um ber R in [0,1] and assign it to F ix).

S tep 2: Solve f o rx from R ~ F ix ) .

The above sam pling m ethod is know n as m ethod o f inversion.

S a m p lin g fr o m e x p o n e n tia l d istrib u tio n

For an exponential distribution the p d f is f i t ) = lie '* 11 fi > 0 t> 0

= 0 r < 0 .

Then

F (t)= ( ' n e - ^ d x = l - e'*1'.

(

6

⁾

(2) I f the random n um ber draw n is R then R = F ( t )

or, R = ^ - e - ^ l. ,

or, f = ~ l j n ( l - i ? ) = - £ - l n K . (7)

T h e last step is possible because if R is a random n u m b e r o n [0, 1] then so is (1 - R) and w e can replace (1 - R ) b y R fo r convenience.

It h a s been established before that if the failure p ro c e ss is d escrib ed by a Poisson distribution, then the tim e b e tw e e n the occurrence o f failures (inter-failure tim e ) m u st follow th e co rresponding exponential d istrib u tio n . T h u s in order to sim ulate the com ponent fa ilu re p ro ce ss d escribed by the P oisson distribution with m e an lit, o v er a tim e period [0, T], all one has to do is to sa m p le th e co rresp o n d in g exponential distribution with m ean 1 //x as m any tim es as necessary until the sum o f the c o rre sp o n d in g exponential random sam ples generated e x c e e d s T for th e first tim e. It can further be explained as follow s.

S u p p o se R t is the ith random sam ple draw n from u n ifo rm [0, 1], then

ti = - j r \ n R i ( 8)

is th e ith sa m p le from the exponential distribution (3).

T h e re fo re T i= Z tj

j= I

(9)

/,= X WyOj

⁽¹⁰⁾

w ith Oj as the outp u t o f the y'th neuron in the previous layer and w(/ is the co nnection w eight betw een the ith node o f one layer and the y'th node o f the previous layer.

T he output o f a node i is obtained as

(11) w here / i s the activation function 11!. M ostly the activation function is sigm oidal w ith th e form

f i x ) = (12)

gives th e tim e instant w hen the ith (com ponent) failure occu rs. T he p ro cess is repeated fo r the m axim um num ber o f tim es (K, say) such that Tk < T.

MAKING THE LEARNING SYSTEMS ROBUST

T h o u g h th e m ethodology th at is g oing to be d eveloped fo r m ak in g neural n etw ork based learning system s ro b u st is true for any N N based system s, the presen t d isc u ssio n is m ade by co n sid erin g a specific type o f N N (the m u lti-lay e r perceptron). S o, let us briefly describ e th e arc h itec tu re and w o rk in g principles o f the m u lti-lay e r (5erceptron first.

In g e n e ra l, a m ultilayer p erceptron (M L P ) M is m ade up o f sets o f n o d es arranged in layers. N odes o f tw o differen t co n sec u tiv e la y ers are co n n e cted b y links or w eights, but th e re is no con n ectio n a m o n g th e elem ents o f the sa m e layer. T h e lay er w here the in p u ts are presented is know n as th e in p u t layer. O n th e o th e r h and th e output producing la y er is ca lle d the o u tp u t layer. T h e layers in betw een the in p u t and th e o u tp u t layers are know n as hidddn layers. T h e o utput o f n o d es in one layer is transm itted to n o d es in ano th er la y er via links that am plify o r a tten u a te o r in h ib it su ch o utputs through w eighting factors. E x ce p t fo r the in put lay er nodes, the total in p u t to e a ch n o d e is th e sum o f w eighted outputs o f the nodes in th e p rev io u s layer. E ach n o d e is activated in accordance w ith th e in put to th e n o d e and th e activation function o f th e node. T he total in p u t (/,) to the ith unit o f any lay er is,

T he function is sym m etrical around Q, and 0O controls the steepness o f the function. 6 is know n as the threshold/bias value.

The back-propagation learning algorithm

F or the operation o f the m ulti-layer perceptron, initially very small random values are assigned to the links/w eights. In the learning p hase'(training) o f such a netw ork w e present the pattern X = (x ,), w here*, is the ith com ponent o f the vector X, as input and ask the net to adjust its set o f w eights in the connecting links and also the thresholds in the nodes such that the desired output {r, | is obtained at the o u tp u t nodes. A fter this, we present an o th er pair o f X and {/,}, and ask the net to learn that association also. In fact, w e desire the net to find a sim ple set o f w eights and biases th a t w ill be able to discrim inate am ong all th e in put/output pairs presented to it. T his process can pose a very strenuous learning task and is not alw ays readily accom plished. H ere the desired output basically acts as a teacher w hich tries to m inim ize the error.

In general, the o utput {»,} will not be the sam e as the target o r desired value {f,}. F or a pattern the error is,

£ = 4 - £

* i

(13) w here the factor o f one h a lf is inserted for m athem atical convenience. T h e increm ental change in w eights fo r a particular pattern p is given by [1]

Awji = T}Sj o,

with

dE

(14)

(15) A s E can be directly calculated in the output layer, fo r the links co nnected to the o u tp u t la y er the change in w eig h t is given by

A

^Wji

= 7] <- - ^ - ) / ’

dE ( I j) O i

.

⁽¹⁶⁾

F o r the w eights w hich do not affect the o u tp u t nodes directly

dE

d o j = X k

^(~

⁸

^k)

w

^>kj.

⁽

¹⁷

⁾

H ence

dE

v (L skwkj)f'(fP °i

k

(18)

for th e o u tp u t lay er and othei layers, respectively. In particu lar e q u a tio n s (11) and (12), if

O j = f ( I j ) =

---

⁷⁷

(

¹⁹

)

1

then

do;

( 20 )

and thus w e g et

A Wji =

n ( L sk wkj) °j (! - op (>i ⁽²¹⁾

fo r the output layer and other layers, respectively.

It m ay be m entioned here that a large value o f r]

corresponds to rap id learning but m ight result in oscillations. A m om entum term o f a Wjt (t) can be added to increase th e learn in g rate and thus expression (14) can be m odified as

AWji (t + 1) = jjS j o,- + a Awji (t) (

22

) w here the quantity ( f + 1) is used to indicate the (t + l)th tim e instant, and a is a proportionality constant. The second term is used to specify that the change in at (t + l) th in stan t should b e som ew hat sim ilar to th e change undertaken a t in stan t t.

Making the learning systems robust

In any system , com ponents may fail with passage o f tim e. In c a se o f N N based system s the com ponents are the neu ro n s/n o d es an d links. So, in such system s som e o f the neurons o r links or b oth m ay get dam aged in .c o u rse of tim e. N N based inform ation processing system s are norm ally c laim ed to b e ro b u st under com ponents failure as th e N N arc h itectu res, involve m assive processing elem ents and connectivity am ong them (m ostly with red u n d a n t co m p o n en ts). T hus even if we dam age som e o f th e co m p o n en ts in the learning phase, the strength o f other links an d b ia ses o f oth er nodes will autom atically get adjusted so as to co m p en sate for this dam age d u rin g the

rest p art o f learning resulting in h ig h er classification accuracy during testing phase. T his basic property can be used to design robust learning system s.

L et T b e the total tim e required fo r / iterations to learn the param eters o f a.N N on a m onoprocess system (w hich can roughly be estim ated from previous experim ents).

Then the tim e required per iteration is

r=z

I '

(

²³

)

N ote that tim e spent for testing is neg lig ib le com pared to T (learning tim e). A lso let t be the tim e req u ired fo r updating a single node (includes collecting the input to it, transform ing the input to output, updating the links connected to it). In practical case, t m ay n ot b e equal fo r all nodes. S uppose there is an n-layer netw ork; w here the operation betw een layers is strictly sequential and operation am ong the nodes in th e sam e layer are parallel.

L et there be nj nodes in the first layer; n 2 nodes in the second layer, « 3 nodes in the third layer and so on. As no tim e is spent in the input layer,

So,

t - ■ T

n2 + «3 + •••+ n„

(

²⁴

)

(25)

Since there are (n - 1) layers w hich o p e ra te ' strictly sequentially, the tim e required' p er ite ratio n fo r parallel im plem entation is

( « - ! ) / = •

( n - l ) r

n2 + n3 + ...+ nn (26)

T hus total tim e (for / iterations) req u ired fo r parallel im plem entation is

T„ = - ( n - l ) r

- / = • 0n - l ) T

(27)

If D com ponents are to be d am ag e d d u rin g this process, then the param eter (ji) for the P oisson distribution (1) is estim ated as,

D

(

²⁸

)

L et the inter-dam age tim e periods (sam ples) be tp j = 1,2, - ., L (L = D). N ow fo r each o f the L tim e instants select a com ponent to b e d am aged. In other words, select L com ponents and d am ag e the ith selected com ponent after T, seconds, w here

7 = 1 i • (29)

N ow if the ith com ponent is to be d am ag e d a t a tim e

Tj-, a n d T, falls in fcth iteration, then for the fcth and su b se q u e n t iterations assum e the ith com ponent as d am ag e d .

T h u s w e n o tice that the com ponents are getting d a m ag e d at various m om ents o f learning. O nly one c o m p o n e n t m ay g et dam aged at the end o f the learning phase. S in ce th e learning process continues even after d a m a g in g o f so m e o f the com ponents, th e adjustm ent o f th e o th e r co m p o n en ts will be in a w ay so as to get the o p tim u m p erfo rm an ce (i.e., with this configuration only th e e rro r value is m inim ized) thereby com pensating for th is d am ag e. N o w if these com ponents w ould have been failed afte r th e co m p letio n o f learning, the perform ance o f th e sy stem w ould d egrade a lot. S ince in a N N based sy stem failu re o f com ponents is natural, it is better to in c o rp o ra te th is fact (by deliberately assum ing som e o f th e c o m p o n en ts as dam aged) w hile learning the p ara m ete rs th ereb y achieving robust .perform ance. Please n o te th a t even th o u g h there is no ch a n g e in the learning alg o rith m , inco rp o ratio n o f dam aging o f com ponents d u rin g learn in g w ill m ake the system robust.

A sim ila r d iscu ssio n can be m ade on treating the links as se p arate co m p o n en ts and assum ing th at the total tim e is sp e n t o n u p d atin g th e links.

W e have co n d u c te d the present sim ulation study by d a m ag in g th e links only. This is due to the fact that the n u m b e r o f nodes in th e hidden layer is very sm all and the

usefulness o f the p roposed algorithm can n o t be d em onstrated rigorously. A s th ere rem ains no redundancy in the input and output nodes the present technique m ay not be useful to study the failure process o f these nodes. A sim ilar study can be d one by d am ag in g a com bination o f links and nodes.

RESULTS AND DISCUSSION

T h e proposed m ethod is im plem ented and tested on the standard M L P based classification problem . Im plem entation is d one on th e IRIS data set (150 sam ples) w hich has 4 input features and 3 classes. The netw ork architecture chosen is 4-3-3, i.e., it has 4 input nodes, one hidden layer w ith 3 nodes and 3 nodes in the output layer. For different sim ulations 10%, 20% and 50%

sam ples w ere taken random ly for training; and the w hole set (o f 150) was taken for testing. The percentage o f co rrect classification (w ith different training sizes and param eter values) are depicted in Tables 1-4. Sim ilar experim ents w ere perform ed by dam aging 2 links and 4 links (o u t o f 4 x 3 + 3 x 3 = 21) w hile the network is in th e training phase. T he classification accuracies are also p u t in th e sam e set o f tables. T he dam aging o f the links w ere p erform ed according to the procedure described earlier. A n o th er se t o f experim ents w ere perform ed using th e learned netw ork by dam aging the sam e set o f links (i.e., those links w hich w ere dam aged during training p hase o f th e form erly described set o f experim ents) in the

T A B LE 1 Percentage o f correct classification w ith learning ra te tj= 0.2 and m om entum value a =0.2

Training Usual Damaged during learning Damaged during testing

size (in %) MLP 2-components 4-components 2-components 4-components

10 96 96 66 96 66

20 96 96 % 96 92

50 98 66 64 66 33

T A B LE 2 Percentage o f correct classification w ith learning r a te t; = 0.2 and m om entum value a =0.5

Training Usual Damaged during learning Damaged during testing

size, (in %) MLP 2-components 4-components 2-components 4-components

10 96 98 94 33 33

20 95 96 62 93 33

50 98 98 66 98 66

T A B L E 3 Percentage o f correct classification w ith learning ra te tj= 0.5 and m om entum value a =: 0.2

Training Usual Damaged during learning Damaged during testing

size (in %) MLP 2-components 4-components 2-components 4-components

10 98 96 96 97 95

20 98 94 94 33 . 33

50 97 97 97 33 33

TA BLE 4 Percentage of correct classification with learning rate tj = 0.5 and m omentum value a = 0.5

Training Usual Damaged during learning Damaged during testing

size (in %) MLP 2-components 4-components 2-components 4-components

10 93 93 93 95 95

20 98 98 96 62 33

50 96 96 96 66 33

testing phase an d e v a lu a tin g the classification accuracy on this dam aged arc h itec tu re.T h e percentage scores obtained by dam aging 2 an d 4 links (o f this settled netw ork) are also included in T ables (1 - 4).

"From th e ta b les w e notice that in m ost o f the cases (with the sam e n etw o rk architecture and sam e set o f param eter v alu es) th e classification accuracy is m ore when the links w ere dam aged during the training/learning phase than they w ere dam aged during testing. For exam ple let us co n sid e r the situation with 77 = 0.5 and a = 0.2 (Table 3). F or 20% training sam ple, the classification accuracy is 98% w ithout any dam age. T he accuracy reduces to 94% with dam aging 2 and 4 links during learning. B ut, if th e sam e links are d am aged w hile testing (on the learned network) the classification accuracy is drastically reduced to 33% . T h u s it is advisable to co n sid er th e possibility o f com p o n en t failure in NN w hile learning its param eters thereby m aking the system more robust.

CONCLUSIONS AND FURTHER SUGGESTIONS

A m ethod to m ak e neural netw ork based learning system s robust w ith respect to com p o n en t failure (dam aging o f n o des/links) is suggested in th e present investigation. T he co m p o n en ts are allowed to be dam aged at different instants o f the learning phase; thereby allow ing tim e to the new architecture (w ith few er com ponents) to adjust its param eters so as to provide optim um p erform ance. T hus th e overall perform ance will n ot be d eteriorated m uch due to this dam age. The dam ag in g /co m p o n en t failure process has been m odeled as a PoissOn process. T h e instants or m om ents o f dam aging are chosen by statistical sam pling. The com ponents to be dam aged are d eterm in e d random ly. As an illustration, the p roposed m odel is im p lem en ted and tested on th e back- propagation le arn in g b a s e d , classification algorithm . A com parative stu d y o f th e scores obtained by the proposed learning system and the standard back-propagation algorithm estab lish es the superiority o f the proposed algorithm .

The w ork p rese n ted here show s a prelim inary attem p t on desig n in g N N b ased ro b u st learning system s. A num ber o f p ro b lem s re la te d to this contribution rem a in s to be investigated. T h e m o s t im p o rtan t and natural extension o f the present c o n c e p t w ill be to develop algorithm s w hich can handle sy stem s w ith p artially dam aged com ponents.

F uzzy set theoretic approach seem s to be a viable alternative for this task. F urther, g en e raliz atio n o f this m odel so as to h andle neuro-fuzzy learn in g system s w hich deal with linguistic or fuzzy input vecto rs and provide output with m ultiple class labels and certainty factors w ill constitute another im portant study.

ACKNOWLEDGEMENT

A part o f this w ork was d one w hen D r A shish G hosh held a research fellow ship o f the M inistry o f E ducation, Science, Sports and C ulture, Govt, o f Japan.

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AUTHOR

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Ashish Ghosh is a Lecturer at the Machine Intelligence Unit, Indian Statistical Institute, Calcutta. He received the BE degree in Electronics and Telecommunication from the Jadavpur University, Calcutta in 1987, and the MTech and PhD degrees in Computer Science from the Indian Statistical Institute,

Calcutta in 1989 and 1993, respectively. He received the prestigious and most coveted Young Scientists award in Engineering Sciences from the Indian National Science

Academy in 1995; and in Computer Science from the Indian Science Congress Association in 1992. He has been selected as an Associate of the Indian Academy of Sciences, Bangalore in 1997. He visited the Osaka Prefecture University, Japan with a Post-doctoral fellowship during October 1995 to March 1997; and Hannan University, Japan as a visiting scholar during September-October 1997. His research interests include Evolutionary Computation, Neural Networks, Image Processing, Fuzzy Sets and Systems, and Pattern Recognition.