Pattern Classification using Simplified Neural Networks with Pruning Algorithm

Transcription

1 Pattern Classification using Siplified Neural Networks with Pruning Algorith S. M. Karuzzaan 1 Ahed Ryadh Hasan 2 Abstract: In recent years, any neural network odels have been proposed for pattern classification, function approxiation and regression probles. This paper presents an approach for classifying patterns fro siplified NNs. Although the predictive accuracy of ANNs is often higher than that of other ethods or huan experts, it is often said that ANNs are practically black boxes, due to the coplexity of the networks. In this paper, we have an attepted to open up these black boxes by reducing the coplexity of the network. The factor akes this possible is the pruning algorith. By eliinating redundant weights, redundant input and hidden units are identified and reoved fro the network. Using the pruning algorith, we have been able to prune networks such that only a few input units, hidden units and connections left yield a siplified network. Experiental results on several bencharks probles in neural networks show the effectiveness of the proposed approach with good generalization ability. Keywords: Artificial Neural Network, Pattern Classification, Pruning Algorith, Weight Eliination, Penalty Function, Network Siplification. 1 Introduction In recent years, any neural network odels have been proposed for pattern classification, function approxiation and regression probles [2] [3] [18]. Aong the, the class of ulti-layer feed forward networks is ost popular. Methods using standard back propagation perfor gradient descent only in the weight space of a network with fixed topology [13]. In general, this approach is useful only when the network architecture is chosen correctly [9]. Too sall a network cannot learn the proble well or too large a size will lead to over fitting and poor generalization [1]. Artificial neural networks are considered as efficient coputing odels and as the universal approxiators [4]. The predictive accuracy of neural network is higher than that of other ethods or huan experts, it is generally difficult to understand how the network arrives at a particular decision due to the coplexity of a particular architecture [6] [15]. One of the ajor criticis is their being black boxes, since no satisfactory explanation of their behavior has been offered. This is because of the coplexity of the interconnections between layers and the network size [18]. As such, an optial network size with inial nuber of interconnection will give insight into how neural network perfors. Another otivation for network siplification and pruning is related to tie coplexity of learning tie [7] [8]. 2 Pruning Algorith Network pruning offers another approach for dynaically deterining an appropriate network topology. Pruning techniques [11] begin by training a larger than necessary network and then eliinate weights and neurons that are deeed redundant. Typically, ethods for reoving weights involve adding a penalty ter to the error function [5]. It is hoped that adding a penalty ter to the error function, unnecessary connection will have saller weights and therefore coplexity of the network can be significantly reduced. This paper ais at pruning the 1 Assistant Professor, Departent of Coputer Science and Engineering, Manarat International University, Dhaka- 1212, Bangladesh, Eail: sk_iiuc@yahoo.co 2 School of Counication, Independent University Bangladesh, Chittagong, Bangladesh. Eail: ryadh78@yahoo.co

2 network size both in nuber of neurons and nuber of interconnections between the neurons. The pruning strategies along with the penalty function are described in the subsequent sections. 2.1 Penalty Function When a network is to be pruned, it is a coon practice to add a penalty ter to the error function during training [16]. Usually, the penalty ter, as suggested in different literature, is 2 2 h n β ( w h o h n h o l ) β ( v ) p 2 2 P( w, v) = ε ε ( wl ) + ( vp ) 1 l 11 β ( w ) 1 p 11 ( ) = = 1 l 1 1 p 1 l = = β v (1) + + p = = = = Given an n-diensional exaple i x, iε { 1, 2,..., k} as input, let wl be the weight for the connection fro input unit ll, ε { 1, 2,..., n} to hidden unit, ε { 1, 2,..., h} unit to output unit p, pε { 1, 2,..., o} and v p be the weight for the connection fro hidden, the p th output of the network for exaple x i is obtained by coputing h i Sp = σ α vp, where = 1 (2) n i α = δ xlwl, ( ) ( x )/ ( δ x = e e e e x ) l= 1 (3) i The target output fro an exaple x that belongs to class C j is an o-diensional vector t i, where t i p = 0 if p = j i and t p = 1, j, p = 1,2 o. The back propagation algorith is applied to update the weights (w, v) and iniize the following function: θ wv, = F wv, + P wv, (4) ( ) ( ) ( ) where F( w, v ) is the cross entropy function as defined k o i i i i (, ) = ( plog p + (1 p)log(1 p) ) F w v t S t S (5) i= 1 p= 1 and P( w, v ) is a penalty ter as described in (1) used for weight decay. 2.2 Redundant weight Pruning Penalty function is used for weight decay. As such we can eliinate redundant weights with the following Weight Eliination Algorith as suggested in different literature [12][14][17] Weight Eliination Algorith: 1. Let η 1 and η 2 be positive scalars such that η 1 + η 2 < Pick a fully connected network and train this network such that error condition is satisfied by all input patterns. Let (w, v) be the weights of this network. w l 3. For each, if v w l 4 η 2 (6) Then reove wl fro the network

3 4. For each v, If v 4 η 2 (7) Then reove v fro the network 5. If no weight satisfies condition (6) or condition (7) then reove v w l. wl with the sallest product 6. Retrain the network. If classification rate of the network falls below an acceptable level, then stop. Otherwise go to Step Input and Hidden Node Pruning A node-pruning algorith is presented below to reove redundant nodes in the input and hidden layer Input and Hidden node Pruning Algorith: Step 1: Create an initial network with as any input neurons as required by the specific proble description and with one hidden unit. Randoly initialize the connection weights of the network within a certain range. Step 2: Partially train the network on the training set for a certain nuber of training epochs using a training algorith. The nuber of training epochs, τ, is specified by the user. Step 3: Eliinate the redundant weights by using weight eliination algorith as described in section 2.2. Step 4: Test this network. If the accuracy of this network falls below an acceptable range then add one ore hidden unit and go to step 2. Step 5: If there is any input node x l with w l = 0, for = 1,2 h, then reove this node. Step 6: Test the generalization ability of the network with test set. If the network successfully converges then erinate, otherwise, go to step Experiental Results And Discussions In this experient, we have used three benchark classification probles. The probles are breast cancer diagnosis, classification of glass types and Pia Indians Diabetes diagnosis proble [10] [19]. All the data sets were obtained fro the UCI achine learning benchark repository. Brief characteristics of the data sets are listed in Table 1. Table 1: Characteristics of data sets. Data set Input Output Output Training Validation Test Total Attributes Units Classes Exaples Exaples exaples exaples Cancer Glass Diabetes The experiental results of different data sets are shown in table 2, figure 1, 2 and 3. In the experiental results of cancer data set, we have found that a fully connected network of architecture has the classification accuracy of

4 97.143%. After pruning the network with Weight Eliination Algorith and Input and Hidden node Pruning Algorith, we have found a siplified network of architecture with classification accuracy of %. The graphical representation of the siplified network is given in figure 3. It shows that only the input attributes I 1, I 6, I 9 along with a single hidden unit is adequate for this proble. Output Layer Hidden Layer O 1 O 2 W 1 = W 6 = W 9 = V 1 = V 2 = Input Layer I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 W i = Input to Hidden Weight V i = Hidden to Output Weight I i = Input signal O i = Output Signal Active Weight Pruned Weight Active Neuron Pruned Neuron Figure 1: Siplified Network for Breast Cancer Diagnosis proble. O 1 O 2 W12 = W 13 = Output Layer W 14 = W 18 = W 21 = W 23 = Hidden Layer W 24 = W 25 = W 26 = W 27 = Input Layer W 28 = I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 V 11 = V 12 = V 21 = V 22 = W i = Input to Hidden Weight V i = Hidden to Output Weight Active Weight Pruned Weight I i = Input signal Active Neuron O i = Output Signal Pruned Neuron Figure 2: Siplified Network for Pia Indians Diabetes diagnosis proble.

5 In the experiental results of Pia Indians Diabetes data set, we have found that a fully connected network of architecture has the classification accuracy of %. After pruning the network with Weight Eliination Algorith and Input and Hidden node Pruning Algorith, we have found a siplified network of architecture with classification accuracy of %. The graphical representation of the siplified network is given in figure 2. It shows that no input attribute can be reoved but a hidden node along with soe redundant connection has been reoved which have been shown with a dotted line in figure 2. O 1 O 2 O 3 O 4 O 5 O 6 Output Layer Hidden Layer Input Layer I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I i = Input signal O i = Output Signal Active Weight Pruned Weight Active Neuron Pruned Neuron Figure 3: Siplified Network for Glass classification proble. In the experiental results of Glass classification data set, we have found that a fully connected network of architecture has the classification accuracy of %. After pruning the network with Weight Eliination Algorith and Input and Hidden node Pruning Algorith, we have found a siplified network of architecture with classification accuracy of %. The graphical representation of the siplified network is given in figure 3. It shows that no input attribute can be reoved but a hidden node along with soe redundant connection has been reoved which have been shown with a dotted line in figure 3.

6 Table 2: Experiental Results Data sets Cancer1 Diabetes Glass Results Learning Rate No. of Epoch Initial Architecture Input Nodes Reoved Hidden Nodes Reoved Total Connection Reoved Siplified Architecture Accuracy (%) of fully connected network Accuracy (%) of siplified network Fro the experiental results discussed above, it can be said that not all input attributes and weights are equally iportant. Moreover, it is difficult to deterine the appropriate nuber of hidden nodes. By pruning approach we can autoatically deterine an appropriate nuber of hidden nodes. We can reove redundant nodes and connections without sacrificing significant accuracy using network pruning approach discussed in section 2.2 and section 2.3. As such we can reduce coputational cost by using the siplified networks. 4 Future Work In future we will use this network pruning approach for rule extraction and feature selection. These pruning strategies will be also exained for function approxiation and regression probles. 5 Conclusions In this paper we proposed an efficient network siplification algorith using pruning strategies. Using this approach we obtain optial network architecture with inial nuber of connections and neurons without deteriorating the perforance of the network significantly. Experiental results show that the perforance of the siplified network is quite significant and acceptable copared to fully connected network. This siplification of the network ensures both reliability and reduced coputational cost. Reference [1] T. Ash, Dynaic node creation in backpropagation networks, Connection Sci., vol. 1, pp , [2] R. W. Brause, Medical Analysis and Diagnosis by Neural Networks, J.W. Goethe-University, Coputer Science Dept., Frankfurt a. M., Gerany. [3] J. W., Everhart, J. E., Dickson, W. C., Knowler, W. C., Johannes, R. S., Using the ADAP learning algorith to forecast the onset of diabetes ellitus, Proc. Syp. on Coputer Applications and Medical Care (Piscataway, NJ: IEEE Coputer Society Press), pp , 1988.

7 [4] S. E. Fahlan and C. Lebiere, The cascade-correlation learning architecture, in Advances in Neural Inforation Processing Syste 2, D. S. Touretzky, Ed. San Mateo, CA: Morgan Kaufann, pp , [5] Sion Haykin, Neural Networks- A Coprehensive Foundation, Second Edition, Pearson Edition Asia, Third Indian Reprint, [6] T. Y. Kwok and D. Y. Yeung, Constructive Algorith for Structure Learning in feed- forward neural network for regression probles, IEEE Trans. Neural Networks, vol. 8, pp , [7] M. Monirul. Isla and K. Murase, A new algorith to design copact two hidden-layer artificial neural networks, Neural Networks, vol. 4, pp , [8] M. Monirul Isla, M. A. H. Akhand, M. Abdur Rahan and K. Murase, Weight Freezing to Reduce Training Tie in Designing Artificial neural Networks, Proceedings of 5 th ICCIT, EWU, pp , Deceber [9] R. Parekh, J.Yang, and V. Honavar, Constructive Neural Network Learning Algoriths for Pattern Classification, IEEE Trans. Neural Networks, vol. 11, no. 2, March [10] L. Prechelt, Proben1-A Set of Neural Network Benchark Probles and Bencharking Rules, University of Karlsruhe, Gerany, [11] R. Reed, Pruning algoriths-a survey, IEEE Trans. Neural Networks, vol. 4, pp , [12] R. Setiono and L.C.K. Hui, Use of quasi-newton ethod in a feedforward neural network construction algorith, IEEE Trans. Neural Networks, vol. 6, no.1, pp , Jan [13] R. Setiono, Huan Liu, Understanding Neural networks via Rule Extraction, In Proceedings of the International Joint conference on Artificial Intelligence, pp , [14] R. Setiono, Huan Liu, Iproving Backpropagation Learning with Feature Selection, Applied Intelligence, vol. 6, no. 2, pp , [15] R. Setiono, Extracting rules fro pruned networks for breast cancer diagnosis, Artificial Intelligence in Machine, vol. 8, no. 1, pp , [16] R. Setiono, A penalty function approach for pruning Feedforward neural networks, Neural Coputation, vol. 9, no. 1, pp , [17] R. Setiono, Techniques for extracting rules fro artificial neural networks, Plenary Lecture presented at the 5th International Conference on Soft Coputing and Inforation Systes, Iizuka, Japan, October [18] R. Setiono W. K. Leow and J. M. Zurada, Extraction of rules fro artificial neural networks for nonlinear regression, IEEE Trans. Neural Networks, vol. 13, no.3, pp , [19] W. H. Wolberg and O.L. Mangasarian, Multisurface ethod of pattern separation for edical diagnosis applied to breast cytology, Proceedings of the National Acadey of Sciences, U SA, Volue 87, pp , Deceber 1990.