Batch Gradient Method for Training of Pi-Sigma Neural Network With Penalty

8202019 Batch Gradient Method for Training of Pi-Sigma Neural Network With Penalty

httpslidepdfcomreaderfullbatch-gradient-method-for-training-of-pi-sigma-neural-network-with-penalty 110

International Journal of Artificial Intelligence amp Applications (IJAIA) Vol 7 No 1 January 2016

DOI 105121ijaia20167102 11

B ATCH GRADIENT METHOD FOR TRAINING OF

PI-SIGMA NEURAL NETWORK WITH PENALTY

Kh Sh Mohamed12

Yan Liu3 Wei Wu

1 Habtamu Z A

1

1School of Mathematical Sciences Dalian University of Technology Dalian 116024

China2Mathematical Department College of Sciences Dalanj University Dalanj Sudan

3School of Information Science and Engineering Dalian Polytechnic University Dalian

116034 China

A BSTRACT

In this letter we describe a convergence of batch gradient method with a penalty condition term for anarration feed forward neural network called pi-sigma neural network which employ product cells as the

output units to inexplicit amalgamate the capabilities of higher-order neural networks while using a minimal number of weights and processing units As a rule the penalty term is condition proportional to

the norm of the weights The monotonicity of the error function with the penalty condition term in the

training iteration is firstly proved and the weight sequence is uniformly bounded The algorithm is applied

to carry out for 4-dimensional parity problem and Gabor function problem to support our theoretical

findings

K EYWORDS

Pi-sigma Neural Network Batch Gradient Method Penaltyamp Convergence

1 INTRODUCTION

Introduced another higher order feed forward polynomial neural network called the pi-sigmaneural network (PSN)[2] which is known to provide naturally more strongly mapping abilities

than traditional feed forward neural network The neural networks consisting of the PSN moduleshas been used effectively in pattern classification and approximation problems [171013] Thereare two ways of training to updating weight The first track batch training the weights are

updating after each training pattern is presented to the network in [9] Second track onlinetraining the weights updating immediately after each training sample is fed (see [3]) The penalty

condition term is oftentimes inserted into the network training algorithms has been vastly used soas to amelioration the generalization performance which refers to the capacity of a neuralnetwork to give correct outputs for untrained data and to control the magnitude of the weights of

the network structure [5612] In the second track the online training weights updating become

very large and over-fitting resort to occur by joining the penalty term into the error function[481114] which acts as a brute-force to drive dispensable weights to zero and to prevent the

weights from taking too large in the training process The objective of this letter to prove thestrong and weak convergence main results which are based on network algorithm prove that the

weight sequence generated is uniformly bounded

For related work we mention [15] where a sigma-pi-sigma network is considered The pi-sigmanetwork considered in this paper has different structure as sigma-pi-sigma network and leads to




12

different theoretical analysis But some techniques of proofs in [15] are also used here in thispaper

The rest of this paper is organized as follows The neural network structure and the batch gradientmethod with penalty is described in the Section 2 In Section 3 the main convergence results are

presented Simulation results are provided in Section 4 In Section 5 the proofs of the main results

are provided Finally some conclusions are drawn in Section 6

2 BATCH GRADIENT METHOD WITH A PENALTY TERM

In this paper we are concerned with a PSNs with the structure p-n-1 where p n and 1 are the

dimensions of the input hidden and output layers respectively Let = hellip isinℝ1 le le the weight vectors connecting the input and summing units and write = hellip isin ℝ Corresponding to the biases with fixed value-1 The structure ofPSN is shown in Fig1

Figure 1 A pi-sigma network with p-n-1 structure

Assume gℝ rarr ℝ is a given activation function In specially for an input isin ℝ the output of

the pi-sigma network is

= g ∙ 1

The network keeping with a given set of training examples 983089

sub ℝ 983255 ℝ is the

numbers of training examples The error function with a penalty is given by

= 12 minus g ∙

+

= g ∙ + 2

where gt 983088 is a penalty coefficient and gt = 983089

983090 minus g 983090 The gradient of with

respect to is written as




13

nabla = g ∙ ∙ +

2

3

Then the weights updating rule is

= minus ∆ = 01hellip 4 ∆ = minusnabla

= gprime ∙ ∙ +

2

5

denotes th update and gt 0 is the learning rate In this paper we suppose that is a fixed

constant and ∙ denotes the Euclidean norm

3 MAIN RESULTS

In this section we present some convergence theorems of the batch gradient method with penalty

(4) These proofs are given in next section Some sufficient conditions for the convergence are as

follows

(A1) g gprime gprimeprime le forall isin ℝ 1 le le

(A2) le C ∙ le forall1 le le 1 le le = 01hellip

(A3) and are chosen to satisfy the condition 0 lt lt

(A4) There exists a closed bounded region Ω such sub Ω and the set Ω = 983164 983101 983088

contains only finite points

Theorem 1 If Assumptions (A1) ndash (A3) are valid let the error function is given by (2) and the

weight sequence be generated by the iteration algorithm (4) for an arbitrary initial valuethen we have

le 983101 983088983089983090 991270983086 There exists lowast ge 983088 such that l983145983149rarrinfin = lowast limrarrinfin

= 0 = 12 Furthermore if Assumption (A4) is also valid then we have the following strong convergence There exists a point lowast isin Ω such that limrarrinfin = lowast

The monotonicity and limit of the error function sequence are shown in Statements

and respectively Statements (ii) and indicate the convergence of

referred to as weak convergence The strong convergence of is described in Statement

4 SIMULATIONS RESULTS

To expound the convergence of batch gradient method for training pi-sigma neural networknumerical example experiments are executed for 4-parity problem and regression problem




14

41 Example 1 Parity Problem

Parity problem is a difficult classification problem The famous XOR problem is completely thetwo-parity problem In this example we use the four-parity problem to test the performance of

PSNs The network is of three layers with the structure 5-4-1 and the logistic activation functiong = 11 + is used for the hidden and output nodes The initial weights are chosen in

[minus05 05] and the learning rate with different value = 005 007 and 009 and the penalty

parameter = 00001 The maximum number of epoch 3000

From Figures 2(a) (b) and 3(c) we observe that the error function and gradient of norm decrease

monotonically respectively and that both norm of the gradient error function approaches zero asdepicted by the convergence Theorem 1 and from Figures Figure 3(d) (e) and (f) we can see the

that the valid function approximation

42 Example 2 Function Regression Problem

In this section we test the performance of batch gradient with penalty for a multi-dimensional

Gabor function has the following form (see Figure 5)

= 983089

983090983088983093 983090 + 983090

983090983088983093 983090 983090 + In this example 256 input points are selected from an evenly 983089983094 983255 983089983094 grid on minus983088983093 le le 983088 983093 and minus983088983093 le le 983088983093 and the 16 input points are randomly selected from the 256 points as training

patterns The number of neurons for input summation and product layer are p=3 N=6 and 1

respectively The parameters in this example take the values = 983088983097 and = 983088 9830889830880983089 when the

number of training iteration epochs reaches 30000(a) (b)

Figure 2 Example 1 (a) Error function with penalty (b) Norm of gradient with penalty

0 500 1000 1500 2000 2500 300010

-6

10-4

10-2

100

102

Number of Iterations

e r r o r f u n c t i o n

η= 005 λ=00001

η= 007 λ=00001

η= 009 λ=00001

0 500 1000 1500 2000 2500 300010

-4

10-3

10-2

10-1

100


N

o r m

o f G

r a d i e n t

η= 005 λ=00001

η= 007 λ=00001

η= 009 λ=00001




15

(c) (d)

Figure 3 Example 2 (c) Error function with penalty (d) Gabor function

(e) (f)

Figure 3 The approximation (e) training pattern results (f) test pattern results

5 PROOFS

To proof Theorem 1 First we present a important lemma which contribute to the our analysis

which is basically the same as Theorem 1415 in [16] Their proof is thus omitted

Lemma 1 Suppose that

ℎ ℝ

rarr ℝ is continuous and differentiable on a compact set

sub ℝ

and that Ω = isin 983164nablaℎ 983101 983088 has only finite number of point If a sequence infin

isin satisfies then limrarrinfin minus = 0 limrarrinfinnablaℎ = 0 Then there exists a pointlowast isin Ω such that limrarrinfin

= lowast

0 05 1 15 2 25 3

x 104

10-2

10-1

100

101

number of iterations


η= 05 λ=00001

η= 07 λ=00001

η= 09 λ=00001

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1




16

Proof of Theorem 1 For sake of convenience we show the notations

= ∆

6

= ∆ minus ∆ 7

Proof Applying Taylorrsquos formula to extend g ∙ at ∙ we have

g ∙ = g ∙

+gprime ∙ ∙

minus

+ 12 gprimeprime

∙

minus ∙

+ 12

minus minus

8

Where isin ℝ is on the line segment between ∙ and ∙ After

dealing with (8) by accumulation g ∙ for 1 le le we obtain from (2) (4)

and Taylorrsquos formula we have

= g ∙

+

= minus 1 ∆ + 2 C ∆

+ ∆ + ∆ 9

Where

∆= 12 gprime ∙

∆∆ 10

∆= 1

2 g primeprime

∙

minus ∙

11

∆= 1 ∆ ∙ 12

It follows from (A1) (A2) (5) and Taylorrsquos formula to first and second orders we obtain

∆ le 12 ∆ ∙ ∆




17

le 12 minus 1 ∆

le C

∆

13

Where C = minus 1 By Assumption (A2) (A2) and Cauchy- Schwartz inequality we

have

∙ minus ∙

le ∙

minus + ∙ ∙

983255 minus + ⋯ + ∙

minus le ∆

le ∆ 14

Where = 1 le le 1 le le = 012 hellip Similarly we get

∙

minus ∙

le ∆ 15

When = By Assumptions (A1) (14) and Cauchy Schwartz inequality we obtain

∆ le 12 ∙ minus ∙

le 12 CC ∆

16

By Assumption (A1) (A2) (7) (14) and (15) for = 01 ⋯ we have

∆ le ∆

∙

le +1 + +1 + 2 ∆

le C ∆ 17




18

Where C = +1 + +1 + 2 Transfer (13) (16) and (17) into (9) there holds

minus le minus 1 minus 2 C minus C minus C minus 12 CC ∆

le minus 1 minus C ∆ le 0 18

This completes the proof to report of the Theorem 1

Proof to of the Theorem 1 From the conclusion we know that the non-negative sequence is monotone But it also bounded below Hence there must exist lowast ge 0 such thatlimrarrinfin = lowast The proof to is thus completed

Proof to of the Theorem 1 It the follows from Assumption (A3) that gt 0 Taking

= minus C

and using (18) we get

le minus le ⋯ le minus

Since gt 0 then we have

le lt infin

Setting rarr infin we have

infin

le 1 lt infin Thus

limrarrinfin

= 0

It follows from (5) and Assumption (A1)

limrarrinfin

∆ = 0 = 12hellip 19 This completes the proof

Proof to of the Theorem 1 Note that the error function defined in (2) is continuous

and differentiable According to (16) Assumption (A4) and Lemma 1 we can easily get the

desired result ie there exists a point lowast isin Ω such that

limrarrinfin

= lowast

This completes the proof to

6 CONCLUSION

Convergence results are decided for the batch gradient method with penalty for training pi-sigma neural network (PSN) The penalty term is a condition proportional to the magnitude

of the weights We prove under moderate conditions if Assumptions (A1) - (A3) hold then




19

that the weights of the networks are deterministically bounded in the learning process With

the help of this conclusion to point strongly purpose if Assumption (A4) is also valid thenwe prove that the suggested algorithm converges with probability one to the set of zeroes of

the gradient of the error function in (2) Resemblance the existing similar convergence

results require the boundedness of the weights as a precondition

ACKNOWLEDGEMENTS

We gratefully acknowledge to thank the anonymous referees for their valuable comments andsuggestions on the revision of this paper

REFERENCES

[1] Hussaina A J amp Liatsisb P (2002)lsquorsquo Recurrent pi-sigma networks for DPCM image

codingrsquorsquoNeurocomputing Vol 55 pp 363-382

[2] Shin Y amp Ghosh J (1991) lsquorsquoThe pi-sigma network An efficient higher-order neural network

forpattern classification and function approximationrsquorsquo International Joint Conference on Neural

Networks Vol 1 pp13ndash18

[3] Wu W amp Xu Y S(2002) lsquorsquo Deterministic convergence of an online gradient method for neural

networksrsquorsquoJournal of Computational and Applied MathematicsVo1 144 (1-2)pp335-347

[4] Geman S Bienenstock E amp Doursat R (1992)lsquorsquoNeural networks and the biasvariance

dilemmarsquorsquoNeural Computation Vo1 4 pp1ndash58

[5] ReedR (1997)lsquorsquoPruning algorithms-a surveyrsquorsquoIEEE Transactions on Neural NetworksVo1 8 pp185ndash

204

[6] G Hinton G (1989) lsquorsquoConnectionist learning proceduresrsquorsquoArtificial Intelligence Vo140 pp185-243

[7] Sinha M Kumar K ampKalra PK (2000) lsquorsquo Some new neural network architectures with improved

learning schemesrsquorsquo Soft Computing Vo14 pp214-223

[8] Setiono R (1997) lsquorsquoA penalty-function approach for pruning feedforward neural networksrsquorsquoNeural

Networks Vo19 pp185ndash204

[9] Wu W Feng G R ampLi X Z (2002)lsquorsquoTraining multiple perceptrons via minimization of sum of

ridgefunctionsrsquo rsquoAdvances in Computational Mathematics Vo117 pp331-347

[10] Shin Yamp Ghosh J(1992)lsquorsquoApproximation of multivariate functions using ridge polynomial

networksrsquorsquo International Joint Conference on Neural Networks Vo1 2 pp380-385[11] Bartlett P Llsquorsquo For valid generalization the size of the weights is more important than the size of the

networkrsquorsquo Advances in Neural Information Processing Systems Vo1 9 pp134ndash140

[12] Loone S amp Irwin G(2001)lsquorsquoImproving neural network training solutions using regularisation

Neurocomputing Vo1 37 pp71-90

[13] Jiang L J Xu F amp Piao S R (2005) lsquorsquoApplication of pi-sigma neural network to real-time

classification of seafloor sedimentsrsquorsquo Applied Acoustics Vo1 24 pp346ndash350

[14] Zhang HS amp Wu W (2009) lsquorsquoBoundedness and convergence of online gradient method with penalty

for linear output feed forward neural networksrsquorsquo Neural Process Letters Vo1 29 pp205ndash212

[15] Liu Y Li Z X Yang DK Mohamed Kh Sh Wang J amp Wu W (2015) lsquorsquo Convergence of batch

gradient learning algorithm with smoothing L12 regularization for Sigma-Pi-Sigma neural

networksrsquorsquo Neurocmputing Vol 151 pp333-341[16] Yuan Y amp Sun W (2001) lsquorsquoOptimization Theory and Methodsrsquorsquo Science Press Beijing




20

Authors

Kh Sh Mohamed received the MS degree from Jilin University Changchun

China in applied mathematics in 2011 He works as a lecturer of mathematics at

College of Science Dalanj University since 2011 Now he is working toward PhD

degree in computational mathematics at Dalian University of Technology DalianChina His research interests include theoretical analysis and regularization methods

for neural networks

Yan Liu received the BS degree in computational mathematics from the Dalian

University of Technology in 2004 She is currently with the School of Information

Sciences and Engineering Dalian Polytechnic University DalianChina In2012 she

has obtained the PhD degree in computational mathematics from Dalian University

of Technology Her research interests include machine learning fuzzy neural

networks and regularization theory

Wei Wu received the Bachelors and Masters degrees from Jilin University

ChangchunChinain1974 and1981 respectively and the PhD degree from Oxford

University OxfordUKin1987He is currently with the School of Mathematical

Sciences Dalian University of Technology Dalian China He has published four

books and 90 research papers His current research interests include learning

methods of neural networks

Habtamu ZA received the bachelor degree from Wollega University Ethiopia in

mathematics education in 2009 and his Masterrsquos degree in Mathematics education

from Addis Ababa University Ethiopia in 2011 He worked as a lecturer of applied

mathematics at Assosa University Ethiopia Currently he is working toward PhDdegree in Applied mathematics at Dalian University of Technology Dalian China

His research interests include numerical optimization methods and neural networks




12

different theoretical analysis But some techniques of proofs in [15] are also used here in thispaper

The rest of this paper is organized as follows The neural network structure and the batch gradientmethod with penalty is described in the Section 2 In Section 3 the main convergence results are

presented Simulation results are provided in Section 4 In Section 5 the proofs of the main results

are provided Finally some conclusions are drawn in Section 6

2 BATCH GRADIENT METHOD WITH A PENALTY TERM

In this paper we are concerned with a PSNs with the structure p-n-1 where p n and 1 are the

dimensions of the input hidden and output layers respectively Let = hellip isinℝ1 le le the weight vectors connecting the input and summing units and write = hellip isin ℝ Corresponding to the biases with fixed value-1 The structure ofPSN is shown in Fig1

Figure 1 A pi-sigma network with p-n-1 structure

Assume gℝ rarr ℝ is a given activation function In specially for an input isin ℝ the output of

the pi-sigma network is

= g ∙ 1

The network keeping with a given set of training examples 983089

sub ℝ 983255 ℝ is the

numbers of training examples The error function with a penalty is given by

= 12 minus g ∙

+

= g ∙ + 2

where gt 983088 is a penalty coefficient and gt = 983089

983090 minus g 983090 The gradient of with

respect to is written as




13

nabla = g ∙ ∙ +

2

3



= gprime ∙ ∙ +

2

5



3 MAIN RESULTS



follows


















14












= 983089

983090983088983093 983090 + 983090






0 500 1000 1500 2000 2500 300010

-6

10-4

10-2

100

102



η= 005 λ=00001

η= 007 λ=00001

η= 009 λ=00001

0 500 1000 1500 2000 2500 300010

-4

10-3

10-2

10-1

100


N

o r m

o f G

r a d i e n t

η= 005 λ=00001

η= 007 λ=00001

η= 009 λ=00001




15

(c) (d)


(e) (f)


5 PROOFS




ℎ ℝ


sub ℝ



= lowast

0 05 1 15 2 25 3

x 104

10-2

10-1

100

101



η= 05 λ=00001

η= 07 λ=00001

η= 09 λ=00001

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1




16


= ∆

6

= ∆ minus ∆ 7


g ∙ = g ∙

+gprime ∙ ∙

minus

+ 12 gprimeprime

∙

minus ∙

+ 12

minus minus

8




= g ∙

+

= minus 1 ∆ + 2 C ∆

+ ∆ + ∆ 9

Where

∆= 12 gprime ∙

∆∆ 10

∆= 1

2 g primeprime

∙

minus ∙

11

∆= 1 ∆ ∙ 12


∆ le 12 ∆ ∙ ∆




17

le 12 minus 1 ∆

le C

∆

13


have

∙ minus ∙

le ∙

minus + ∙ ∙

983255 minus + ⋯ + ∙

minus le ∆

le ∆ 14


∙

minus ∙

le ∆ 15



le 12 CC ∆

16


∆ le ∆

∙

le +1 + +1 + 2 ∆

le C ∆ 17




18







= minus C




le lt infin


infin

le 1 lt infin Thus

limrarrinfin

= 0


limrarrinfin





limrarrinfin

= lowast


6 CONCLUSION






19





ACKNOWLEDGEMENTS


REFERENCES











204























20

Authors





for neural networks





















13

nabla = g ∙ ∙ +

2

3



= gprime ∙ ∙ +

2

5



3 MAIN RESULTS



follows


















14












= 983089

983090983088983093 983090 + 983090






0 500 1000 1500 2000 2500 300010

-6

10-4

10-2

100

102



η= 005 λ=00001

η= 007 λ=00001

η= 009 λ=00001

0 500 1000 1500 2000 2500 300010

-4

10-3

10-2

10-1

100


N

o r m

o f G

r a d i e n t

η= 005 λ=00001

η= 007 λ=00001

η= 009 λ=00001




15

(c) (d)


(e) (f)


5 PROOFS




ℎ ℝ


sub ℝ



= lowast

0 05 1 15 2 25 3

x 104

10-2

10-1

100

101



η= 05 λ=00001

η= 07 λ=00001

η= 09 λ=00001

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1




16


= ∆

6

= ∆ minus ∆ 7


g ∙ = g ∙

+gprime ∙ ∙

minus

+ 12 gprimeprime

∙

minus ∙

+ 12

minus minus

8




= g ∙

+

= minus 1 ∆ + 2 C ∆

+ ∆ + ∆ 9

Where

∆= 12 gprime ∙

∆∆ 10

∆= 1

2 g primeprime

∙

minus ∙

11

∆= 1 ∆ ∙ 12


∆ le 12 ∆ ∙ ∆




17

le 12 minus 1 ∆

le C

∆

13


have

∙ minus ∙

le ∙

minus + ∙ ∙

983255 minus + ⋯ + ∙

minus le ∆

le ∆ 14


∙

minus ∙

le ∆ 15



le 12 CC ∆

16


∆ le ∆

∙

le +1 + +1 + 2 ∆

le C ∆ 17




18







= minus C




le lt infin


infin

le 1 lt infin Thus

limrarrinfin

= 0


limrarrinfin





limrarrinfin

= lowast


6 CONCLUSION






19





ACKNOWLEDGEMENTS


REFERENCES











204























20

Authors





for neural networks





















14












= 983089

983090983088983093 983090 + 983090






0 500 1000 1500 2000 2500 300010

-6

10-4

10-2

100

102



η= 005 λ=00001

η= 007 λ=00001

η= 009 λ=00001

0 500 1000 1500 2000 2500 300010

-4

10-3

10-2

10-1

100


N

o r m

o f G

r a d i e n t

η= 005 λ=00001

η= 007 λ=00001

η= 009 λ=00001




15

(c) (d)


(e) (f)


5 PROOFS




ℎ ℝ


sub ℝ



= lowast

0 05 1 15 2 25 3

x 104

10-2

10-1

100

101



η= 05 λ=00001

η= 07 λ=00001

η= 09 λ=00001

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1




16


= ∆

6

= ∆ minus ∆ 7


g ∙ = g ∙

+gprime ∙ ∙

minus

+ 12 gprimeprime

∙

minus ∙

+ 12

minus minus

8




= g ∙

+

= minus 1 ∆ + 2 C ∆

+ ∆ + ∆ 9

Where

∆= 12 gprime ∙

∆∆ 10

∆= 1

2 g primeprime

∙

minus ∙

11

∆= 1 ∆ ∙ 12


∆ le 12 ∆ ∙ ∆




17

le 12 minus 1 ∆

le C

∆

13


have

∙ minus ∙

le ∙

minus + ∙ ∙

983255 minus + ⋯ + ∙

minus le ∆

le ∆ 14


∙

minus ∙

le ∆ 15



le 12 CC ∆

16


∆ le ∆

∙

le +1 + +1 + 2 ∆

le C ∆ 17




18







= minus C




le lt infin


infin

le 1 lt infin Thus

limrarrinfin

= 0


limrarrinfin





limrarrinfin

= lowast


6 CONCLUSION






19





ACKNOWLEDGEMENTS


REFERENCES











204























20

Authors





for neural networks





















15

(c) (d)


(e) (f)


5 PROOFS




ℎ ℝ


sub ℝ



= lowast

0 05 1 15 2 25 3

x 104

10-2

10-1

100

101



η= 05 λ=00001

η= 07 λ=00001

η= 09 λ=00001

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1

-05

0

05

-05

0

05

-05

0

05

1




16


= ∆

6

= ∆ minus ∆ 7


g ∙ = g ∙

+gprime ∙ ∙

minus

+ 12 gprimeprime

∙

minus ∙

+ 12

minus minus

8




= g ∙

+

= minus 1 ∆ + 2 C ∆

+ ∆ + ∆ 9

Where

∆= 12 gprime ∙

∆∆ 10

∆= 1

2 g primeprime

∙

minus ∙

11

∆= 1 ∆ ∙ 12


∆ le 12 ∆ ∙ ∆




17

le 12 minus 1 ∆

le C

∆

13


have

∙ minus ∙

le ∙

minus + ∙ ∙

983255 minus + ⋯ + ∙

minus le ∆

le ∆ 14


∙

minus ∙

le ∆ 15



le 12 CC ∆

16


∆ le ∆

∙

le +1 + +1 + 2 ∆

le C ∆ 17




18







= minus C




le lt infin


infin

le 1 lt infin Thus

limrarrinfin

= 0


limrarrinfin





limrarrinfin

= lowast


6 CONCLUSION






19





ACKNOWLEDGEMENTS


REFERENCES











204























20

Authors





for neural networks





















16


= ∆

6

= ∆ minus ∆ 7


g ∙ = g ∙

+gprime ∙ ∙

minus

+ 12 gprimeprime

∙

minus ∙

+ 12

minus minus

8




= g ∙

+

= minus 1 ∆ + 2 C ∆

+ ∆ + ∆ 9

Where

∆= 12 gprime ∙

∆∆ 10

∆= 1

2 g primeprime

∙

minus ∙

11

∆= 1 ∆ ∙ 12


∆ le 12 ∆ ∙ ∆




17

le 12 minus 1 ∆

le C

∆

13


have

∙ minus ∙

le ∙

minus + ∙ ∙

983255 minus + ⋯ + ∙

minus le ∆

le ∆ 14


∙

minus ∙

le ∆ 15



le 12 CC ∆

16


∆ le ∆

∙

le +1 + +1 + 2 ∆

le C ∆ 17




18







= minus C




le lt infin


infin

le 1 lt infin Thus

limrarrinfin

= 0


limrarrinfin





limrarrinfin

= lowast


6 CONCLUSION






19





ACKNOWLEDGEMENTS


REFERENCES











204























20

Authors





for neural networks





















17

le 12 minus 1 ∆

le C

∆

13


have

∙ minus ∙

le ∙

minus + ∙ ∙

983255 minus + ⋯ + ∙

minus le ∆

le ∆ 14


∙

minus ∙

le ∆ 15



le 12 CC ∆

16


∆ le ∆

∙

le +1 + +1 + 2 ∆

le C ∆ 17




18







= minus C




le lt infin


infin

le 1 lt infin Thus

limrarrinfin

= 0


limrarrinfin





limrarrinfin

= lowast


6 CONCLUSION






19





ACKNOWLEDGEMENTS


REFERENCES











204























20

Authors





for neural networks





















18







= minus C




le lt infin


infin

le 1 lt infin Thus

limrarrinfin

= 0


limrarrinfin





limrarrinfin

= lowast


6 CONCLUSION






19





ACKNOWLEDGEMENTS


REFERENCES











204























20

Authors





for neural networks





















19





ACKNOWLEDGEMENTS


REFERENCES











204























20

Authors





for neural networks





















20

Authors





for neural networks


















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Batch Gradient Method for Training of Pi-Sigma Neural Network With Penalty

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