A Brief Overview of Neural Networks

Overview• Relation to Biological Brain: Biological Neural Network• The Artificial Neuron• Types of Networks and Learning Techniques• Supervised Learning & Backpropagation Training

Algorithm• Learning by Example• Applications

Biological Neuron

Artificial Neuron

Σ f(n)W

W

W

W

Outputs

Activation

Function

INPUTS

W=Weight

Neuron

Transfer Functions

: ( ) 11 nSIGMOID f n

e

: ( )LINEAR f n n

1

0 Input

Output

Types of networks

Multiple Inputs and Single Layer

Multiple Inputs and layers

Types of Networks – Contd.Feedback

Recurrent Networks

Recurrent Networks• Feed forward networks:

– Information only flows one way– One input pattern produces one output– No sense of time (or memory of previous state)Recurrency– Nodes connect back to other nodes or

themselves– Information flow is multidirectional– Sense of time and memory of previous state(s)

• Biological nervous systems show high levels of recurrency (but feed-forward structures exists too)

ANNs – The basics

• ANNs incorporate the two fundamental components of biological neural nets:

1. Neurones (nodes)

2. Synapses (weights)

Feed-forward nets

• Information flow is unidirectional• Data is presented to Input layer

• Passed on to Hidden Layer

• Passed on to Output layer

• Information is distributed

• Information processing is parallel

Internal representation (interpretation) of data

Neural networks are good for prediction problems.

• The inputs are well understood. You have a good idea of which features of the data are important, but not necessarily how to combine them.

• The output is well understood. You know what you are trying to predict.

• Experience is available. You have plenty of examples where both the inputs and the output are known. This experience will be used to train the network.

• Feeding data through the net:

(1 0.25) + (0.5 (-1.5)) = 0.25 + (-0.75) = - 0.5 0.3775

11

5.0

eSquashing:

Learning Techniques

• Supervised Learning:

Inputs from the environment

Neural Network

Actual System

Σ

Error

+

-

Expected Output

Actual Output

Training

Multilayer Perceptron

Inputs First Hidden layer

Second Hidden Layer

Output Layer

Signal FlowBackpropagation of Errors

Function SignalsError Signals

Neural networks for Directed Data Mining: Building a model for classification and prediction

1. Identify the input and output features2. Normalize (scaling) the inputs and outputs so their range is

between 0 and 1.3. Set up a network on a representative set of training examples.4. Train the network on a representative set of training examples.5. Test the network on a test set strictly independent from the

training examples. If necessary repeat the training, adjusting the training set, network topology, nad parameters. Evaluate the network using the evaluation set to see how well it performs.

6. Apply the model generated by the network to predict outcomes for unknown inputs.

Learning by Example

• Hidden layer transfer function: Sigmoid function = F(n)= 1/(1+exp(-n)), where n is the net input to the neuron.

Derivative= F’(n) = (output of the neuron)(1-output of the neuron) : Slope of the transfer function.

• Output layer transfer function: Linear function= F(n)=n; Output=Input to the neuron

Derivative= F’(n)= 1

Purpose of the Activation Function

• We want the unit to be “active” (near +1) when the “right” inputs are given

• We want the unit to be “inactive” (near 0) when the “wrong” inputs are given.

• It’s preferable for activation function to be nonlinear. Otherwise, the entire neural network collapses into a simple linear function.

Possibilities for activation function

Step function Sign function Sigmoid (logistic) function

step(x) = 1, if x > threshold 0, if x threshold(in picture above, threshold = 0)

sign(x) = +1, if x > 0 -1, if x 0

sigmoid(x) = 1/(1+e-x)

Adding an extra input with activation a0 = - 1 and weightW0,j = t (called the bias weight) is equivalent to having a threshold at t. This way we can always assume a 0 threshold.

Using a Bias Weight to Standardize the Threshold

-1T

x1

x2

W1

W2

W1x1+ W2x2 < T

W1x1+ W2x2 - T < 0

Real vs artificial neurons

axon

dendrites

dendrites

synapsecell

x0

xn

w0

wn

oi

n

iixw

0

otherwise 0 and 0 if 10

i

n

iixwo

Threshold units

Implementing AND

x1

x2

o(x1,x2)

otherwise 0 05.1 if 1),( 2121

xxxxo

1

1

-1

W=1.5

Assume Boolean (0/1) input values…

Implementing OR

x1

x2

o(x1,x2)

1

1

-1

W=0.5

o(x1,x2) = 1 if –0.5 + x1 + x2 > 0 = 0 otherwise

Implementing NOT

x1 o(x1,x2)-1

W=-0.5-1

otherwise 0 05.0 if 1)( 11

xxo

Implementing more complex Boolean functions

x1

x2

11

0.5-1

x1 or x2

x3

11

1.5

(x1 or x2) and x3

-1

Learning by Example

• Training Algorithm: backpropagation of errors using gradient descent training.

• Colors:– Red: Current weights– Orange: Updated weights– Black boxes: Inputs and outputs to a neuron– Blue: Sensitivities at each layer

• The perceptron learning rule performs gradient descent in weight space.– Error surface: The surface that describes the error on

each example as a function of all the weights in the network. A set of weights defines a point on this surface. (It could also be called a state in the state space of possible weights, i.e., weight space.)

– We look at the partial derivative of the surface with respect to each weight (i.e., the gradient -- how much the error would change if we made a small change in each weight). Then the weights are being altered in an amount proportional to the slope in each direction (corresponding to a weight). Thus the network as a whole is moving in the direction of steepest descent on the error surface.

Definition of Error: Sum of Squared Errors

22

21)(

21 ErrotE

examples

Here, t is the correct (desired) output and o is the actualoutput of the neural net.

This is introduced to simplify the math on the next slide

Reduction of Squared ErrorGradient descent reduces the squared error by calculatingthe partial derivative of E with respect to each weight:

j

n

kkk

j

jj

xingErr

xWgtW

Err

WErrErr

WEE

)('0

jjj xingErrWW )('

chain rule for derivatives

expand second Err above to (t – g(in))

This is called “in”

0

jWt

because and chain rule

The weight is updated by η times this gradient of error E in weight space. The fact that the weight is updated in the correct direction (+/-) can be verified with examples.

learning rate

The learning rate, η, is typically set to a small value such as 0.1

E is a vector

First Pass

0.5

0.5

0.5

0.50.50.5

0.5

0.51

0.5

0.5 0.6225

0.62250.6225

0.6225

0.6508

0.6508

0.6508

0.6508

Error=1-0.6508=0.3492

G3=(1)(0.3492)=0.3492

G2= (0.6508)(1-0.6508)(0.3492)(0.5)=0.0397

G1= (0.6225)(1-0.6225)(0.0397)(0.5)(2)=0.0093

Gradient of the neuron= G =slope of the transfer function×[Σ{(weight of the neuron to the next neuron) × (output of the neuron)}]

Gradient of the output neuron = slope of the transfer function × error

Weight Update 1New Weight=Old Weight + {(learning rate)(gradient)(prior output)}

0.5+(0.5)(0.3492)(0.6508)

0.6136

0.5124 0.5124

0.51240.6136

0.5124

0.5047

0.5047

0.5+(0.5)(0.0397)(0.6225)0.5+(0.5)(0.0093)(1)

Second Pass

0.5047

0.5124

0.6136

0.61360.50470.5124

0.5124

0.51241

0.5047

0.50470.6391

0.63910.6236

0.6236

0.8033

0.6545

0.6545

0.8033

Error=1-0.8033=0.1967

G3=(1)(0.1967)=0.1967

G2= (0.6545)(1-0.6545)(0.1967)(0.6136)=0.0273

G1= (0.6236)(1-0.6236)(0.5124)(0.0273)(2)=0.0066

Weight Update 2New Weight=Old Weight + {(learning rate)(gradient)(prior output)}

0.6136+(0.5)(0.1967)(0.6545)

0.6779

0.5209 0.5209

0.52090.6779

0.5209

0.508

0.508

0.5124+(0.5)(0.0273)(0.6236)0.5047+(0.5)(0.0066)(1)

Third Pass

0.508

0.5209

0.6779

0.67790.5080.5209

0.5209

0.52091

0.508

0.5080.6504

0.65040.6243

0.62430.8909

0.6571

0.6571

0.8909

Weight Update Summary

Output Expected OutputErrorw1 w2 w3

Initial conditions 0.5 0.5 0.5 0.6508 1 0.3492Pass 1 Update 0.5047 0.5124 0.6136 0.8033 1 0.1967Pass 2 Update 0.508 0.5209 0.6779 0.8909 1 0.1091

Weights

W1: Weights from the input to the input layerW2: Weights from the input layer to the hidden layerW3: Weights from the hidden layer to the output layer

Training Algorithm

• The process of feedforward and backpropagation continues until the required mean squared error has been reached.

• Typical mse: 1e-5• Other complicated backpropagation

training algorithms also available.

Why Gradient?O1

O2

O = Output of the neuronW = Weight N = Net input to the neuron

W1

W2 N = (O1×W1)+(O2×W2)

O3 = 1/[1+exp(-N)]

Error = Actual Output – O3

• To reduce error: Change in weights: o Learning rateo Rate of change of error w.r.t rate of change of weight

Gradient: rate of change of error w.r.t rate of change of ‘N’ Prior output (O1 and O2)

0 Input

Output 1

Gradient in Detail• Gradient : Rate of change of error w.r.t rate of change in net input to neuron

o For output neurons Slope of the transfer function × error

o For hidden neurons : A bit complicated ! : error fed back in terms of gradient of successive neurons

Slope of the transfer function × [Σ (gradient of next neuron × weight connecting the neuron to the next neuron)] Why summation? Share the responsibility!!

An Example

1

0.4

0.731

0.598

0.5

0.50.5

0.5

0.6645

0.6645

0.66

0.66

1

0

Error = 1-0.66 = 0.34

Error = 0-0.66 = -0.66

G1=0.66×(1-0.66)×(-0.66)= -0.148

G1=0.66×(1-0.66)×(0.34)= 0.0763

Reduce more

Increase less

Improving performance

• Changing the number of layers and number of neurons in each layer.

• Variation in Transfer functions.• Changing the learning rate. • Training for longer times.• Type of pre-processing and post-

processing.

Applications

• Used in complex function approximations, feature extraction & classification, and optimization & control problems

• Applicability in all areas of science and technology.

Neural networks with more than one output

– A department store chain wants to predict the likelihood that customers will be purchasing products from various departments, like women apparel, furniture, and entertainment.

– The store wants to use this information to plan promotions and direct target mailings.

– This network has three outputs, one for each department. The outputs are a propensity for the customer described in the inputs to make his or her next purchase from the associated department.

gender

age

Last purchase Propensity to purchase Women’s apparel

Propensity to purchase furniture

Propensity to purchase entertainment

How can the department store determine the right promotion or promotions to offer the customer?• Taking the department corresponding to the unit

with the maximum value• Taking the departments corresponding to the

units with the top three values• Taking the departments corresponding to the

units that exceed some threshold value• Taking all departments corresponding to units

that are some percentage of the unit with he maximum value.