Chapter 3 Neural Network Xiu-jun GONG (Ph. D) School of Computer Science and Technology, Tianjin University

Outline  Introduction  Training a single TLU  Network of TLUs—Artificial Neural Network  Pros & Cons of ANN  Summary

Biological /Artificial Neural Network SMI32-stained pyramidal neurons in cerebral cortex. Structure of a typical neuron x 2 w 2 wnwn … x 1 w 1  f(s) F(s) xnxn Artificial Intelligence Recognition modeling Neuroscience

Definition of ANN  Stimulate Neural Network: SNN, NN It is an interconnected group of artificial neurons that uses a mathematical or computational model for information processing based on a connectionist approach to computation. In most cases an ANN is an adaptive system that changes its structure based on external or internal information that flows through the network.

Applications of ANN  Function approximation, or regression analysis, including time series prediction and modeling.  Classification, including pattern and sequence recognition, novelty detection and sequential decision making.  Data processing, including filtering, clustering, blind signal separation and compression.

Extension of a TLU  Threshold Logic Unit -> Perceptron (Neuron) Inputs are not limited be boolean values Outputs are not limited be binary functions

Output functions of a perceptron θ

Characters of sigmoid function  Smooth, continuous, and monotonically increasing (derivative is always positive)  Bounded range - but never reaches max or min  The logistic function is often used

Linear Separable function by TLU

A network of TLUs x1x1 x2x2 y1y1 y2y2 f XOR Even-Parity Function

Training single neuron  What is the learning/training  The methods The Delta Procedure The Generalized Delta Procedure The Error-Correction Procedure

Reform the representation of a perceptron x1x2…xnx1x2…xn x n+1 ≡ 1 W1W1 W2W2 WnWn W n+` f = f (s) S=WXS=WX Summing Junction Activation Function output

Gradient Decent Methods  Minimizing the squared error of desired response and neuron output Squared error function: ε = (d - f) 2

The Delta Procedure  Using linear function f = s  Weight update: W ← W + c (d – f ) X  Delta rule ( Widrow-Hoff rule)

The Generalized Delta Procedure  Using sigmoid function f (s) = 1 / (1+e -s )  Weight update W ← W + c (d – f ) f (1-f ) X  Generalized delta procedure: f (1– f ) → 0, where f → 0 or f → 1 Weight change can occur only within ‘fuzzy’ region surrounding the hyperplane near the point f = 0.5

The Error-Correction Procedure  Using threshold function (output : 0,1)  The weight change rule W ← W + c (d – f ) X W ← W ± c X  In the linearly separable case, after finite iterations, W will be converged to the solution.  In the nonlinearly separable case, W will never be converged.

An example x1=S2+S3 x2=S4+S5 x3=S6+S7 x4=S8+S9 x1 x2 x3 x4 1 W 11 W 21 W 41 W 31 W 51 east

ANN: Its topologies Context Layer Recurrent ANN Inputs Feedback Outputs Feedforward Inputs Outputs

Training Neural Network  Supervised method Trained by matching input and output patterns Input-output pairs can be provided by an external teacher, or by the system  Unsupervised method (Self-organization) An (output) unit is trained to respond to clusters of pattern within the input. There is no a priori set of categories  Enforcement learning An intermediate form of the above two types of learning. The learning machine does some action on the environment and gets a feedback response from the environment. The learning system grades its action good (rewarding) or bad (punishable) based on the environmental response and accordingly adjusts its parameters.

Supervised training

Back-propagation—Notations |||||| |||||| |||||| |||||| |||||| ----

Back-propagation: The method 1. Initialize connection weights into small random values. 2. Present the p t h sample input vector of pattern and the corresponding output target to the network 3. Pass the input values to the first layer, layer 1. For every input node i in layer 0, perform: 4 For every neuron i in every layer j = 1, 2,..., M, from input to output layer, find the output from the neuron: 5. Obtain output values. For every output node i in layer M, perform: 6.Calculate error value for every neuron i in every layer in backward order j = M, M-1,..., 2, 1

The method cont. 6.1 For the output layer, the error value is: 6.2 For the hidden layer, the error value is: 6.3 The weight adjustment can be done for every connection from neuron k in layer (i-1) to every neuron i in every layer i: The actions in steps 2 through 6 will be repeated for every training sample pattern p, and repeated for these sets until the root mean square (RMS) of output errors is minimized.

Generalization vs. specialization  Optimal number of hidden neurons Too many hidden neurons : you get an over fit, training set is memorized, thus making the network useless on new data sets Not enough hidden neurons: network is unable to learn problem concept  Overtraining: Too much examples, the ANN memorizes the examples instead of the general idea  Generalization vs. specialization trade-off K-fold cross validation is often used

Unsupervised method  No help from the outside  No training data, no information available on the desired output  Learning by doing  Used to pick out structure in the input: Clustering Reduction of dimensionality  compression  Kohonen’s Learning Law (Self- Organization Map) Winner takes all (only update weights of winning neuron)

SOM algorithm

An example: Kohonen Network.

Reinforcement learning  Teacher: training data  The teacher scores the performance of the training examples  Use performance score to shuffle weights ‘randomly’  Relatively slow learning due to ‘randomness’

Anatomy of ANN learning algorithm ANN Learning SupervisedLogic inputsHopfield Continuous inputs Back propagation Unsupervis ed Logic inputsART Continuous inputs SOM, Hebb, Reinforcem ent learning

Pros & Cons of ANN Pros:  A neural network can perform tasks that a linear program can not.  When an element of the neural network fails, it can continue without any problem by their parallel nature.  A neural network learns and does not need to be reprogrammed.  It can be implemented in any application. Cons :  The neural network needs training to operate.  The architecture of a neural network is different from the architecture of microprocessors therefore needs to be emulated.  Requires high processing time for large neural networks.

Summary  The capability of ANN representations  Training a single perceptron  Training neural networks  The ability of Generalization vs. specialization should be memorized