Dr. kiani artificial neural network lecture 1

Semiconductors, BP&A Planning, 2003-01-29 1

o/w 0 and 0 if 10

ii xwo

Threshold units

History spiking neural networks

Vapnik (1990) ---support vector machine

Broomhead & Lowe (1988) ----Radial basis functions (RBF)

Linsker (1988) ----- Informax principle

Rumelhart, Hinton -------- Back-propagation & Williams (1986)

Kohonen(1982) ------ Self-organizing mapsHopfield(1982) ------ Hopfield Networks

Minsky & Papert(1969) ------ Perceptrons

Rosenblatt(1960) ------ Perceptron

Minsky(1954) ------ Neural Networks (PhD Thesis)

Hebb(1949) --------The organization of behaviour

McCulloch & Pitts (1943) -----neural networks and artificial intelligence were born

History of Neural Networks

• 1943: McCullough and Pitts - Modeling the Neuron for Parallel Distributed Processing

• 1958: Rosenblatt - Perceptron• 1969: Minsky and Papert publish limits on

the ability of a perceptron to generalize• 1970’s and 1980’s: ANN renaissance• 1986: Rumelhart, Hinton + Williams

present backpropagation• 1989: Tsividis: Neural Network on a chip

William McCulloch

Neural Networks

• McCulloch & Pitts (1943) are generally recognised as the designers of the first neural network

• Many of their ideas still used today (e.g. many simple units combine to give increased computational power and the idea of a threshold)

Neural Networks

• Hebb (1949) developed the first learning rule (on the premise that if two neurons were active at the same time the strength between them should be increased)

Neural Networks

• During the 50’s and 60’s many researchers worked on the perceptron amidst great excitement.

• 1969 saw the death of neural network research for about 15 years – Minsky & Papert

• Only in the mid 80’s (Parker and LeCun) was interest revived (in fact Werbos discovered algorithm in 1974)

How Does the Brain Work ? (1)

NEURON• The cell that perform information

processing in the brain• Fundamental functional unit of

all nervous system tissue

How Does the Brain Work ? (2)

Each consists of : SOMA, DENDRITES, AXON, and SYNAPSE

Biological neurons

dendrites

synapse

Neural Networks

• We are born with about 100 billion neurons

• A neuron may connect to as many as 100,000 other neurons

Biological inspiration

Dendrites

Soma (cell body)

synapses

axon dendrites

The information transmission happens at the synapses.

The spikes travelling along the axon of the pre-synaptic neuron trigger the release of neurotransmitter substances at the synapse.

The neurotransmitters cause excitation or inhibition in the dendrite of the post-synaptic neuron.

The integration of the excitatory and inhibitory signals may produce spikes in the post-synaptic neuron.

The contribution of the signals depends on the strength of the synaptic connection.

Biological Neurons

• human information processing system consists of brain neuron: basic building block

– cell that communicates information to and from various parts of body

• Simplest model of a neuron: considered as a threshold unit –a processing element (PE)

• Collects inputs & produces output if the sum of the input exceeds an internal threshold value

Artificial Neural Nets (ANNs)

• Many neuron-like PEs units– Input & output units receive and broadcast signals to the

environment, respectively

– Internal units called hidden units since they are not in contact with external environment

– units connected by weighted links (synapses)

• A parallel computation system because– Signals travel independently on weighted channels & units

can update their state in parallel– However, most NNs can be simulated in serial computers

• A directed graph, with labeled edges by weights is typically used to describe the connections among units

Each processing unit has a simple program that: a) computes a weighted sum of the input data it receives from those units which feed into it b) outputs of a single value, which in general is a non-linear function of the weighted sum of the its inputs ---this output then becomes an input to those units into which the original units feeds

activationlevel

A NODE

inputfunctionactivation function

output

input linksoutputlinks

aj Wj,i

ai = g(ini)

g = Activation functions for units

Step function(Linear Threshold Unit)

Sign function Sigmoid function

step(x) = 1, if x >= threshold 0, if x < threshold

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

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

Real vs artificial neurons

dendrites

synapse

o/w 0 and 0 if 10

ii xwo

Threshold units

Artificial neurons

Neurons work by processing information. They receive and provide information in form of spikes.

The McCullogh-Pitts model

Inputs

Outputw2

zHyxwzn

Mathematical representation

The neuron calculates a weighted sum of inputs and compares it to a threshold. If the sum is higher than the threshold, the output is set to 1, otherwise to -1.

Non-linearity

• x1

• x2

• xn

• …

• w1• w2

• …

• wn

1threshold threshold • f

iin wxxxxf

121 if,1),...,,(

otherwise,0

Artificial neurons

Basic Concepts

Definition of a node:

• A node is an element which performs the function

y = fH(∑(wixi) + Wb)fH(x)

Input 0 Input 1 Input n...

W0 W1 Wn

Output

NodeNode

ConnectionConnection

Anatomy of an Artificial Neuron

inputs

h(w0 ,wi , xi )

w0 f : activation function

output

h : combine wi & xi

Simple Perceptron

• Binary logic application• fH(x) = u(x) [linear

threshold]• Wi = random(-1,1)

• Y = u(W0X0 + W1X1 + Wb)

• Now how do we train it?

Input 0 Input 1

Output

• From experience: examples / training data

• Strength of connection between the neurons is stored as a weight-value for the specific connection.

• Learning the solution to a problem = changing the connection weights

Artificial Neuron

An artificial neuron

A physical neuron

Mathematical Representation

f(n)..

Inputs Weights Summation Activation Output

Inputs

Outputw2

y f (net)

A simple perceptron

• It’s a single-unit network• Change the weight by

an amount proportional to the difference between the desired output and the actual output.

Δ Wi = η * (D-Y).Ii

Perceptron Learning Rule

Learning rate Desired output

Actual output

Linear Neurons

•Obviously, the fact that threshold units can only output the values 0 and 1 restricts their applicability to certain problems.

•We can overcome this limitation by eliminating the threshold and simply turning fi into the identity function so that we get:

)(net )( tto ii

•With this kind of neuron, we can build networks with m input neurons and n output neurons that compute a function

f: Rm Rn.

Linear Neurons

•Linear neurons are quite popular and useful for applications such as interpolation.

•However, they have a serious limitation: Each neuron computes a linear function, and therefore the overall network function f: Rm Rn is also linear.

•This means that if an input vector x results in an output vector y, then for any factor the input x will result in the output y.

•Obviously, many interesting functions cannot be realized by networks of linear neurons.

Mathematical Representation

nnfa nnfa )(

( ) na f n e

Gaussian Neurons

•Another type of neurons overcomes this problem by using a Gaussian activation function:

ffii(net(netii(t))(t))

netnetii(t)(t)•-1

1)(net

))(net(

Gaussian Neurons

•Gaussian neurons are able to realize non-linear functions.

•Therefore, networks of Gaussian units are in principle unrestricted with regard to the functions that they can realize.

•The drawback of Gaussian neurons is that we have to make sure that their net input does not exceed 1.

•This adds some difficulty to the learning in Gaussian networks.

Sigmoidal Neurons

•Sigmoidal neurons accept any vectors of real numbers as input, and they output a real number between 0 and 1.

•Sigmoidal neurons are the most common type of artificial neuron, especially in learning networks.

•A network of sigmoidal units with m input neurons and n output neurons realizes a network function f: Rm (0,1)n

Sigmoidal Neurons

•The parameter controls the slope of the sigmoid function, while the parameter controls the horizontal offset of the function in a way similar to the threshold neurons.

ffii(net(netii(t))(t))

netnetii(t)(t)•-1

/))(net(1

1))(net( tii ie

Example: A simple single unit adaptive network

• The network has 2 inputs, and one output. All are binary. The output is – 1 if W0I0 + W1I1 + Wb > 0 – 0 if W0I0 + W1I1 + Wb ≤ 0

• We want it to learn simple OR: output a 1 if either I0 or I1 is 1.

Artificial neurons

The McCullogh-Pitts model:

• spikes are interpreted as spike rates;

• synaptic strength are translated as synaptic weights;

• excitation means positive product between the incoming spike rate and the corresponding synaptic weight;

• inhibition means negative product between the incoming spike rate and the corresponding synaptic weight;

Artificial neurons

Nonlinear generalization of the McCullogh-Pitts neuron:

),( wxfy y is the neuron’s output, x is the vector of inputs, and w is the vector of synaptic weights.

Examples:

sigmoidal neuron

Gaussian neuron

NNs: Dimensions of a Neural Network

– Knowledge about the learning task is given in the form of examples called training examples.

– A NN is specified by:– an architecture: a set of neurons and links connecting

neurons. Each link has a weight, – a neuron model: the information processing unit of the

NN,– a learning algorithm: used for training the NN by

modifying the weights in order to solve the particular learning task correctly on the training examples.

The aim is to obtain a NN that generalizes well, that is, that behaves correctly on new instances of the learning task.

Neural Network Architectures

Many kinds of structures, main distinction made between two classes:

a) feed- forward (a directed acyclic graph (DAG): links are unidirectional, no cycles

b) recurrent: links form arbitrary topologies e.g., Hopfield Networks and Boltzmann machines

Recurrent networks: can be unstable, or oscillate, or exhibit chaoticbehavior e.g., given some input values, can take a long time to compute stable output and learning is made more difficult….However, can implement more complex agent designs and can

modelsystems with state

We will focus more on feed- forward networks

Single Layer Feed-forward

Input layerof

source nodes

Output layerof

neurons

Multi layer feed-forward

Inputlayer

Outputlayer

Hidden Layer

3-4-2 Network

Feed-forward networks:

Advantage: lack of cycles = > computation proceeds uniformly from input units to output units.

-activation from the previous time step plays no part in computation, as it is not fed back to an earlier unit

- simply computes a function of the input values that depends on the weight settings –it has no internal state other than the weights themselves.

- fixed structure and fixed activation function g: thus the functions representable by a feed-forward network are restricted to have acertain parameterized structure

Learning in biological systems

Learning = learning by adaptation

The young animal learns that the green fruits are sour, while the yellowish/reddish ones are sweet. The learning happens by adapting the fruit picking behavior.

At the neural level the learning happens by changing of the synaptic strengths, eliminating some synapses, and building new ones.

Learning as optimisation

The objective of adapting the responses on the basis of the information received from the environment is to achieve a better state. E.g., the animal likes to eat many energy rich, juicy fruits that make its stomach full, and makes it feel happy.

In other words, the objective of learning in biological organisms is to optimise the amount of available resources, happiness, or in general to achieve a closer to optimal state.

Synapse concept

• The synapse resistance to the incoming signal can be changed during a "learning" process [1949]

Hebb’s Rule: If an input of a neuron is repeatedly and

persistently causing the neuron to fire, a metabolic change happens in the synapse of that particular

input to reduce its resistance

Neural Network Learning

• Objective of neural network learning: given a set of examples, find parameter settings that minimize the error.

• Programmer specifies- numbers of units in each layer - connectivity between units,

• Unknowns- connection weights

Supervised Learning in ANNs

•In supervised learning, we train an ANN with a set of vector pairs, so-called exemplars.

•Each pair (x, y) consists of an input vector x and a corresponding output vector y.

•Whenever the network receives input x, we would like it to provide output y.

•The exemplars thus describe the function that we want to “teach” our network.

•Besides learning the exemplars, we would like our network to generalize, that is, give plausible output for inputs that the network had not been trained with.

•There is a tradeoff between a network’s ability to precisely learn the given exemplars and its ability to generalize (i.e., inter- and extrapolate).

•This problem is similar to fitting a function to a given set of data points.

•Let us assume that you want to find a fitting function f:RR for a set of three data points.

•You try to do this with polynomials of degree one (a straight line), two, and nine.

•Obviously, the polynomial of degree 2 provides the most plausible fit.

•f(x)

•deg. 1

•deg. 2

•deg. 9

Overfitting

Overfitted ModelReal Distribution

•The same principle applies to ANNs:

• If an ANN has too few neurons, it may not have enough degrees of freedom to precisely approximate the desired function.

• If an ANN has too many neurons, it will learn the exemplars perfectly, but its additional degrees of freedom may cause it to show implausible behavior for untrained inputs; it then presents poor ability of generalization.

•Unfortunately, there are no known equations that could tell you the optimal size of your network for a given application; you always have to experiment.

Learning in Neural Nets

Learning Tasks

Supervised UnsupervisedData:Labeled examples (input , desired output)

Tasks:classificationpattern recognition regressionNN models:perceptron adalinefeed-forward NN radial basis functionsupport vector machines

Data:Unlabeled examples (different realizations of the input)

Tasks:clusteringcontent addressable memory

NN models:self-organizing maps (SOM)Hopfield networks

Learning Algorithms

Depend on the network architecture:

• Error correcting learning (perceptron)• Delta rule (AdaLine, Backprop)• Competitive Learning (Self Organizing

Perceptrons

• Perceptrons are single-layer feedforward networks• Each output unit is independent of the others• Can assume a single output unit• Activation of the output unit is calculated by:

• O = Step( )

where xj is the activation of input unit j, and we assume an additional weight and input to represent the threshold

j jxjw0

Perceptron

X0 = 1

j jxjw0 > 01 if

-1 otherwise

j jxjw0

Perceptron

Rosenblatt (1958) defined a perceptron to be a machine that learns, using examples, to assign input vectors (samples) to different classes, using linear functions of the inputs

Minsky and Papert (1969) instead describe perceptron as a stochastic gradient-descent algorithm that attempts to linearly separate a set of n-dimensional training data.

Linear Separable

some functions not representable - e.g., (b) not linearly separable

So what can be represented using perceptrons?

and or

Representation theorem: 1 layer feedforward networks canonly represent linearly separable functions. That is,the decision surface separating positive from negativeexamples has to be a plane.

Learning Boolean AND

• No w0, w1, w2 satisfy: (Minsky and Papert, 1969)

Expressive limits of perceptrons

• Can the XOR function be represented by a perceptron

(a network without a hidden layer)?

XOR cannot be represented.

How can perceptrons be designed?

• The Perceptron Learning Theorem (Rosenblatt, 1960): Given enough training examples, there is an algorithm that will learn any linearly separable function.

Theorem 1 (Minsky and Papert, 1969) The perceptron rule converges to weights that correctly classify all training examples provided the given data set represents a function that is linearly separable

The perceptron learning algorithm

• Inputs: training set {(x1,x2,…,xn,t)}• Method

– Randomly initialize weights w(i), -0.5<=i<=0.5– Repeat for several epochs until convergence:

• for each example– Calculate network output o.– Adjust weights:

)( Perceptron training

learning rate error

Why does the method work?

• 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.

– 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.

• The error surface in weight space has a single global minimum and no local minima. Gradient descent is guaranteed to find the global minimum, provided the learning rate is not so big that that you overshoot it.

Multi-layer, feed-forward networks

Perceptrons are rather weak as computing models since they can only learn linearly-separable functions.

Thus, we now focus on multi-layer, feed forward networks of non- linear sigmoid units: i.e.,

g(x) =

Multi-layer feed-forward networks

Multi-layer, feed forward networks extend perceptrons i.e., 1-layernetworks into n-layer by:• Partition units into layers 0 to L such that;

•lowermost layer number, layer 0 indicates the input units

•topmost layer numbered L contains the output units.

•layers numbered 1 to L are the hidden layers

•Connectivity means bottom-up connections only, with no cycles, hence the name"feed-forward" nets

•Input layers transmit input values to hidden layer nodes hence do notperform any computation.

Note: layer number indicates the distance of a node from the inputnodes

Multilayer feed forward network

x0 x1 x2 x3 x4

v1v2 v3

Layer of input units

Layer of hidden units

Layer of output units

Multi-layer feed-forward networks

• Multi-layer feed-forward networks can be trained by back-propagation provided the activation function g is a differentiable function.– Threshold units don’t qualify, but the sigmoid function

• Back-propagation learning is a gradient descent search through the parameter space to minimize the sum-of-squares error.

– Most common algorithm for learning algorithms in multilayer networks

Sigmoid units

Sigmoid unit for g

))(1)(()(

This is g’ (the basis for gradient descent)

Weight updating in backprop • Learning in backprop is similar to learning with perceptrons, i.e.,

– Example inputs are fed to the network.

• If the network computes an output vector that matches the target, nothing is done.

• If there is a difference between output and target (i.e., an error), then the weights are adjusted to reduce this error.

• The key is to assess the blame for the error and divide it among the contributing weights.

• The error term (T - o) is known for the units in the output layer. To adjust the weights between the hidden and the output layer, the gradient descent rule can be applied as done for perceptrons.

• To adjust weights between the input and hidden layer some way of estimating the errors made by the hidden units in needed.

Estimating Error

• Main idea: each hidden node contributes for some fraction of the error in each of the output nodes. – This fraction equals the strength of the connection

(weight) between the hidden node and the output node.

ioutputsi

j nodehidden at error

where is the error at output node i.i

A goal of neural network learning is, given a set of examples, to find parameter settings that minimize the error

Back-propagation Learning

• Inputs: – Network topology: includes all units & their

connections – Some termination criteria – Learning Rate (constant of proportionality

of gradient descent search) – Initial parameter values– A set of classified training data

• Output: Updated parameter values

Back-propagation algorithm for updating weights in a multilayer network

1.Initialize the weights in the network (often randomly) 2.repeat for each example e in the training set do i.O = neural-net-output(network, e) ; forward pass ii.T = teacher output for e iii.Calculate error (T - O) at the output units iv.Compute wj = wj + * Err * Ij for all weights from

hidden layer to output layer;backward pass v.Compute wj = wj + * Err * Ij for all weights from input layer

to hidden layer; backward pass continued vi.Update the weights in the network end 3.until all examples classified correctly or stopping criterion met 4.return(network)

Back-propagation Using Gradient Descent

• Advantages– Relatively simple implementation– Standard method and generally works

• Disadvantages– Slow and inefficient– Can get stuck in local minima resulting

in sub-optimal solutions

Number of training pairs needed?

Difficult question. Depends on the problem, the training examples, andnetwork architecture. However, a good rule of thumb is:

Where W = No. of weights; P = No. of training pairs, e = error rate

For example, for e = 0.1, a net with 80 weights will require 800training patterns to be assured of getting 90% of the test patternscorrect (assuming it got 95% of the training examples correct).

How long should a net be trained?

• The objective is to establish a balance between correct responses for the training patterns and correct responses for new patterns. (a balance between memorization and generalization).

• If you train the net for too long, then you run the risk of overfitting.

• In general, the network is trained until it reaches an acceptable error rate (e.g., 95%)

Implementing Backprop – Design Decisions

1. Choice of r2. Network architecture

a) How many Hidden layers? how many hidden units per a layer?b) How should the units be connected? (e.g., Fully, Partial, using

domain knowledge3. Stopping criterion – when should training stop?

Determining optimal network structure

Weak point of fixed structure networks: poor choice can lead to poor performance

Too small network: model incapable of representing the desired Function

Too big a network: will be able to memorize all examples but forming a large lookup table, but will not generalize well to inputs that have not been seen before.

Thus finding a good network structure is another example of asearch problems.Some approaches to search for a solution for this problem includeGenetic algorithmsBut using GAs is very cpu-intensive.

Learning rate

• Ideally, each weight should have its own learning rate

• As a substitute, each neuron or each layer could have its own rate

• Learning rates should be proportional to the sqrt of the number of inputs to the neuron

Setting the parameter values

• How are the weights initialized?• Do weights change after the presentation of

each pattern or only after all patterns of the training set have been presented?

• How is the value of the learning rate chosen?• When should training stop?• How many hidden layers and how many nodes

in each hidden layer should be chosen to build a feedforward network for a given problem?

• How many patterns should there be in a training set?

• How does one know that the network has learnt something useful?

When should neural nets be used for learning a problem

• If instances are given as attribute-value pairs.– Pre-processing required: Continuous

input values to be scaled in [0-1] range, and discrete values need to converted to Boolean features.

• Noise in training examples.

• If long training time is acceptable.

Neural Networks: Advantages

•Distributed representations •Simple computations

•Robust with respect to noisy data

•Robust with respect to node failure

•Empirically shown to work well for many problem domains

•Parallel processing

Neural Networks: Disadvantages•Training is slow

•Interpretability is hard

•Network topology layouts ad hoc

•Can be hard to debug

•May converge to a local, not global, minimum of error

•May be hard to describe a problem in terms of features with numerical values

Back-propagation Algorithm

yi(n) wji(n)

wj1(n)

wj0(n)=bj (n)

∑ netj(n) f(.)

( ) ( ) ( )

( ) ( ( ))

( ) ( ) ( )

j ji ii

net n w n y n

y n f net n

e n d n y n

Total error )(2

1)( 2 nen

All output neurons

)(1 Average squared error where N=No. of items in the training set

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )j j j

ji j j j ji

e n y n net nn n

w n e n y n net n w n

( )( ) ( ( )) ( )

( ) j j j iji

ne n v n y n

1)( 2 nen

)()()( nyndne jjj

( )' ( ( ))

y nf net n

( ) ( ( ))j j jy n f net n

( )( )

net ny n

( ) ( ) ( )m

j ji ii

net n w n y n

Gradient decent

)()()( nynnw ijji

Error term

where ( ) ( ) '( ( ))j j jn e n f net n

)()()( nynnw ijji 1

( ( ))1 exp( ( ))j j

net nnet n

'( ( )) ( )[1 ( )]j j jnet n y n y n

( ) ( ( ))j j jy n net n

Neuron k is an output node

( ) ( ) '( ( )) [ ( ) ( )] ( )[1 ( )]k k k k k k kn e n net n d n y n y n y n

( ) '( ( )) ( ) ( )

( )[1 ( )] ( ) ( )

j j k kjk

n net n n w n

y n y n n w n

Neuron j is a hidden node

Output layer (k)Hidden layer(j)

)()()( nynnw ijji

Weight adjustment

Learningrate

Localgradient

Input signal

( 1) ( ) ( )ji ji jiw n w n w n

Output of neuron k

Brain vs. Digital Computers (1)

Computers require hundreds of cycles to simulate a firing of a neuron

The brain can fire all the neurons in a single step. ParallelismSerial computers require billions of cycles to

perform some tasks but the brain takes less than a seconde.g. Face Recognition

What are Neural Networks

• An interconnected assembly of simple processing elements, units, neurons or nodes, whose functionality is loosely based on the animal neuron

• The processing ability of the network is stored in the interunit connection strengths, or weights, obtained by a process of adaptation to, or learning from, a set of training patterns.

Definition of Neural Network

A Neural Network is a system composed of manysimple processing elements operating in parallelwhich can acquire, store, and utilize experientialKnowledge.

Architecture

• Connectivity:– fully connected– partially connected

• Feedback– feedforward network: no feedback

• simpler, more stable, proven most useful

– recurrent network: feedback from output to input units

• complex dynamics, may be unstable

• Number of layers i.e. presence of hidden layers

Hidden

Output

Inputs

NodeConnection

Outputs

Feedforward, Fully-Connected with One Hidden Layer

Hidden Units

• Layer of nodes between input and output nodes

• Allow a network to learn non-linear functions

• Allow the net to represent combinations of the input features

Learning Algorithms

• How the network learns the relationship between the inputs and outputs

• Type of algorithm used depends on type of network- architecture, type of learning,etc.

• Back Propagation: most popular– modifications exist: quick prop, Delta-bar-Delta

• Others: Conjugate gradient descent, Levenberg-Marquardt, K-Means, Kohonen, standard pseudo-inverse (SVD) linear optimization

Types of Networks

• Multilayer Perceptron• Radial Basis Function• Kohonen• Linear• Hopfield• Adaline/Madaline• Probabilistic Neural Network (PNN)• General Regression Neural Network

(GRNN) • and at least thirty others

• A Neural Network is a system composed of many simple processing elements operating in parallel which can acquire, store, and utilize experiential knowledge

• Basic Artificial Model– Consists of simple processing elements called neurons, units or

nodes– Each neuron is connected to other nodes with an associated

weight(strength) which typically multiplies the signal transmitted. Each neuron has a single threshold value

• Characterization– Architecture: the pattern of nodes and connections between

them– Learning algorithm, or training method: method for

determining weights of the connections– Activation function: function that produces an output based on

the input values received by node

Perceptrons

• First studied in the late 1950s• Also known as Layered Feed-Forward

Networks• The only efficient learning element at that

time was for single-layered networks• Today, used as a synonym for a single-

layer, feed-forward network

Perceptrons

• First studied in the late 1950s• Also known as Layered Feed-Forward

Networks• The only efficient learning element at that

time was for single-layered networks• Today, used as a synonym for a single-

layer, feed-forward network

Single Layer Perceptron

OUT = F(NET)OUT

X2 • • • Xn

OUT = F(NET)

X2 • • • Xn

Squashing function need not be sigmoidal

Perceptron Architecture

Single-Neuron Perceptron

Decision Boundary

Example OR

OR solution

Multiple-Neuron Perceptron

Learning Rule Test Problem

Starting Point

Tentative Learning Rule

Second Input Vector

Third Input Vector

Unified Learning Rule

Multiple-Neuron Perceptron

Apple / Banana Example

Apple / Banana Example, Second iteration

Apple / Banana Example, Check

Historical Note

There was great interest in Perceptrons in the '50s and '60s - centred on the work of Rosenblatt.

This was crushed by the publication of "Perceptrons" by Minsky and Papert.

• EOR problem

regions must be linearly separable.

• Training Problems

esp. with higher order nets.

Multilayer Perceptron(MLP)

• Type of Back Propagation Network• Arguably the most popular network architecture

today• Can model functions of almost arbitrary

complexity, with number of layers and number of units/layer determining the function complexity

• Number of hidden layers and units: good starting point is to use one hidden layer with the number of units equal to half the sum of the number of input and output units

Perceptrons

Network Structures

Feed-forward:Links can only go in one direction.

Recurrent : Links can go anywhere and form arbitrary

topologies.

Feed-forward Networks

• Arranged in layers• Each unit is linked only in the unit in next layer• No units are linked between the same layer,

back to the previous layer or skipping a layer• Computations can proceed uniformly from input

to output units• No internal state exists

W24= -1

W46 = 1

W67 = 1

W13 = -1

W35 = 1

W57 = 1

W25 = 1

W16 = 1

Feed-Forward Example

Introduction to Backpropagation

• In 1969 a method for learning in multi-layer network, Backpropagation, was invented by Bryson and Ho

• The Backpropagation algorithm is a sensible approach for dividing the contribution of each weight

• Works basically the same as perceptrons

Backpropagation Learning

• There are two differences for the updating rule :– The activation of the hidden unit is used

instead of the input value– The rule contains a term for the gradient of

the activation function

Backpropagation Learning

• There are two differences for the updating rule :– The activation of the hidden unit is used

instead of the input value– The rule contains a term for the gradient of

the activation function

Backpropagation Algorithm Summary

The ideas of the algorithm can be summarized as follows :

• Computes the error term for the output units using the observed error

• From output layer, repeat propagating the error term back to the previous layer and updating the weights between the two layers until the earliest hidden layer is reached

Back Propagation

• Best-known example of a training algorithm• Uses training data to adjust weights and thresholds

of neurons so as to minimize the networks errors of prediction

• Slower than modern second-order algorithms such as gradient descent & Levenberg-Marquardt for many problems

• Still has some advantages over these in some instances

• Easiest algorithm to understand• Heuristic modifications exist: quick propagation and

Delta-bar-Delta

Training a BackPropagation Net

• Feedforward training of input patterns– each input node receives a signal, which is

broadcast to all of the hidden units– each hidden unit computes its activation which is

broadcast to all output nodes

• Back propagation of errors– each output node compares its activation with the

desired output– based on these differences, the error is propagated

back to all previous nodes

• Adjustment of weights– weights of all links computed simultaneously based

on the errors that were propagated back

Training Back Prop Net: Feedforward Stage

1. Initialize weights with small, random values2. While stopping condition is not true

– for each training pair (input/output):• each input unit broadcasts its value to all hidden units• each hidden unit sums its input signals & applies

activation function to compute its output signal• each hidden unit sends its signal to the output units• each output unit sums its input signals & applies its

activation function to compute its output signal

Training Back Prop Net: Backpropagation

3. Each output computes its error term, its own weight correction term and its bias(threshold) correction term & sends it to layer below

4. Each hidden unit sums its delta inputs from above & multiplies by the derivative of its activation function; it also computes its own weight correction term and its bias correction term

Training a Back Prop Net: Adjusting the Weights

5. Each output unit updates its weights and bias6. Each hidden unit updates its weights and bias

– Each training cycle is called an epoch. The weights are updated in each cycle

– It is not analytically possible to determine where the global minimum is. Eventually the algorithm stops in a low point, which may just be a local minimum.

Gradient Descent Methods

• Error Function: How far off are we?– Example Error function:

depends on weight values• Gradient Descent: Minimize error by

moving weights along the decreasing slope of error

• The Idea: iterate through the training set and adjust the weights to minimize the gradient of the error

fdε 2

Gradient Descent: The Math

We have = (d - f)2

Gradient of :

Using the chain rule:

Since , we have

Which finally gives:

,...,,...,ni wwwW

Gradient Descent: Back to reality

• So we have • The problem: f / s is not

differentiable• Three solutions:

– Ignore It: The Error-Correction Procedure– Fudge It: Widrow-Hoff– Approximate it: The Generalized Delta

Procedure

Training a Back Prop Net: Adjusting the Weights

5. Each output unit updates its weights and bias6. Each hidden unit updates its weights and

bias– Each training cycle is called an epoch. The weights

are updated in each cycle– It is not analytically possible to determine where the

global minimum is. Eventually the algorithm stops in a low point, which may just be a local minimum.

How long should you train?

• Goal: balance between correct responses for training patterns & correct responses for new patterns (memorization v. generalization)

• In general, network is trained until it reaches an acceptable error rate (e.g. 95%)

• If train too long, you run the risk of overfitting

Learning in the BPN

•Before the learning process starts, all weights (synapses) in the network are initialized with pseudorandom numbers.

•We also have to provide a set of training patterns (exemplars). They can be described as a set of ordered vector pairs {(x1, y1), (x2, y2), …, (xP, yP)}.

•Then we can start the backpropagation learning algorithm.

•This algorithm iteratively minimizes the network’s error by finding the gradient of the error surface in weight-space and adjusting the weights in the opposite direction (gradient-descent technique).

Learning in the BPN•Gradient-descent example: Finding the absolute minimum of a one-dimensional error function f(x):

slope: f’(x0)

x1 = x0 - f’(x0)

•Repeat this iteratively until for some xi, f’(xi) is sufficiently close to 0.

Learning in the BPN•Gradients of two-dimensional functions:

•The two-dimensional function in the left diagram is represented by contour lines in the right diagram, where arrows indicate the gradient of the function at different locations. Obviously, the gradient is always pointing in the direction of the steepest increase of the function. In order to find the function’s minimum, we should always move against the gradient.

Gradient Descent

(w1,w2)

(w1+w1,w2 +w2)

Learning in the BPN

• In the BPN, learning is performed as follows:

1. Randomly select a vector pair (xp, yp) from the training set and call it (x, y).

2. Use x as input to the BPN and successively compute the outputs of all neurons in the network (bottom-up) until you get the network output o.

3. Compute the error opk, for the pattern p across all K

output layer units by using the formula:

)(')( okkk

opk netfoy

Learning in the BPN

4. Compute the error hpj, for all J hidden layer units by

using the formula:

hpj wnetf

5. Update the connection-weight values to the hidden layer by using the following equation:

ihpjjiji xtwtw )()1(

Learning in the BPN

•Repeat steps 1 to 6 for all vector pairs in the training set; this is called a training epoch.

•Run as many epochs as required to reduce the network error E to fall below a threshold :

6. Update the connection-weight values to the output layer by using the following equation:

)()()1( hj

opkkjkj netftwtw

Learning in the BPN

•The only thing that we need to know before we can start our network is the derivative of our sigmoid function, for example, f’(netk) for the output neurons:

kef k net1

1)net(

)1(net

)net()net(' kk

Learning in the BPN

•Now our BPN is ready to go!

•If we choose the type and number of neurons in our network appropriately, after training the network should show the following behavior:

• If we input any of the training vectors, the network should yield the expected output vector (with some margin of error).

• If we input a vector that the network has never “seen” before, it should be able to generalize and yield a plausible output vector based on its knowledge about similar input vectors.

Weights and Gradient Descent

• Calculate the partial derivatives of the error E with respect to each of the weights:

• Minimize E by gradient descent: change each weight by an amount proportional to the partial derivative

If slope is negative increase wIf slope is positive decrease w

Local minima for E are places wherederivative equals zero

Gradient descent

Local minimum

Global minimum

Faster Convergence: Momentum rule• Add a fraction (=momentum) of the last change to the

current change

)1()( twwEtw

Back-propagation: a simple “chain rule” procedure for updating all weights

output

hidden

Weight updates for hidden to output weights wo are easy to calculate. For sigmoid activation function

oji yw wo

wh Weight updates for input to hidden layer require “back-propagated” quantities

oj yyyd 1

oy1oy2

hy1hy2

hi wyy 1

hjk iw

(a.k.a delta rule)

Procedure for two-layer network (with sigmoid activation functions)

1. Initialize all weights to small random values

2. Choose an input pattern c and set the input nodes to that pattern

3. Propagate signal forward by calculating hidden node activation

4. Propagate signal forward to output nodes

5. Compute deltas for the output node layer

6. Compute deltas for the hidden node layer

7. Update weights immediately after this pattern (no waiting to accumulate weight changes over all patterns)

hi wiy exp11

oj wyy exp11

oj yyyd 1

hi wyy 1

oji yw

hjk iw

Artificial Neural Networks

Perceptrons

o(x1,x2...,xn) = 1 if w0+ w1 x1+.. + wn xn > 0

-1 otherwise

o(x) = sgn(w.x) (x0=1)

Hypothesis Space: H = {w | w n+1}

• Representational Power

– Perceptrons can represent all the primitive Boolean functions AND, OR, NAND (AND) and NOR (OR)

– They cannot represent all Boolean functions (for example, XOR)

– Every Boolean function can be represented by some network of perceptrons two levels deep

• The Perceptron Training Rule

wi wi + wi wi = (t - o) xi

t: target output for the current training example

o: output generated by the perceptron

: learning rate

Multilayer Networks and the BP Algorithm

ANNs with two or more layers are able to represent complex nonlinear decision surfaces

– Differentiable Threshold (Sigmoid) Units

o = (w.x) (y) = 1/(1+e-y)

/y = (y) [1-(y)]

First find the outputs OI , OII . In order to do this, propagate the inputs forward. First find the outputs for the neurons of hidden layer

)1.()(

2321310

2221210

2121110

XWXWWOXWOO

Example: Learning addition

)1.()(

3322110

OWOWOWWOOWOO

IIIIIIi

IIiIIiII

Then find the outputs of the neurons of output layer

Now propagate back the errors. In order to do that first find the errors for the output layer, also update the weights between hidden layer and output layer

IIIIIIIIII

And backpropagate the errors to hidden layer.

))(1()1(

IIIIIIk

WWOOWOO

And backpropagate the errors to hidden layer.

323232

313131

303030

222222

212121

202020

121212

111111

101010

IIIIII

Finally update weights!!!!

Gradient-Descent Training Rule

•As described in Machine Learning (Mitchell, 1997).•Also called delta rule, although not quite the same as the adaline delta rule. •Compute Ei, the error at output node i over the set of training instances, D.

Intuitive: Do what you can to reduce Ei, so: If increases in wij will increase Ei (i.e., dEi/dwij > 0), then reduce wij, but ” ” decrease Ei ” ” < 0, then increase wij

• Compute dEi/dwij (i.e. wij’s contribution to error at node i) for every input weight to node i.

• Gradient Descent Method: Updating all wij by the delta rule amounts to moving along the path of steepest descent on the error surface.

• Difficult part: computing dEi/dwij .

• Base weight updates on dEi/dwij

ididi otE D = training set

= distance * direction (to move in error space)

Computing dEi/dwij

Ddidid

i otww

)()(22

ijDdidid ot

)()( idijDd

idid ow

))(()( idTijDd

idid sumfw

jjdijid xwsumwhere

ij wsum

)( jdid

T xsum

In general:

Computing )( idTij

idsumidT esumf

1)(Sigmoidal ft :

))(1)(()1(1

12 idtidtsum

sumfsumfe

eesum id

ididT osumf )(But since: jdidididTij

xoosumfw

ididT sumsumf )(Identity ft :

sumsum

jdjdij

sumsum

If fT is not continuous, andhence not differentiableeverywhere, then we cannotuse the Delta Rule.

Weight Updates for Simple Units

)()( jdDd

idid xot

ididij

iij xot

))(()( idTijDd

ididij

i sumfw

fT = identity function

wijxjd

Weight Updates for Sigmoidal Units

))(1()( jdididDd

idid xooot

jdididDd

ididij

iij xooot

w )1()(

))(()( idTijDd

ididij

i sumfw

fT = sigmoidal function

wijxjd

Error gradient for the sigmoid function

Bias of a Neuron

• The bias b has the effect of applying an affine transformation to the weighted sum u

v = u + b

x1-x2=0

x1-x2= 1

x2 x1-x2= -1

Bias as extra input• The bias is an external parameter of the neuron. It can be modeled by adding an extra input.

Inputsignal

Synapticweights

Summingfunction

ActivationfunctionLocal

vOutput

w0x0 = +1

Apples/Bananas Sorter

Prototype Vectors

Perceptron

Two-Input Case

Apple / Banana Example

Testing the Network

Questions? Suggestions ?

Dr. kiani artificial neural network lecture 1

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