1. Machine Learning
Neural Networks

2. Introduction
Artificial Neural Network is based on the biological nervous system as Brain
It is composed of interconnected computing units called neurons
ANN like human, learn by examples

3. Why Artificial Neural Networks?
There are two basic reasons why we are interested in building artificial neural networks (ANNs):
Technical viewpoint: Some problems such as character recognition or the prediction of future states of a system require massively parallel and adaptive processing.
Biological viewpoint: ANNs can be used to replicate and simulate components of the human (or animal) brain, thereby giving us insight into natural information processing.

4. Science: Model how biological neural systems, like human brain, work?
How do we see?
How is information stored in/retrieved from memory?
How do you learn to not to touch fire?
How do your eyes adapt to the amount of light in the environment?
Related fields: Neuroscience, Computational Neuroscience, Psychology, Psychophysiology, Cognitive Science, Medicine, Math, Physics.

5. Brief History Old Ages: Association (William James; 1890)
McCulloch-Pitts Neuron (1943,1947)
Perceptrons (Rosenblatt; 1958,1962)
Adaline/LMS (Widrow and Hoff; 1960)
Perceptrons book (Minsky and Papert; 1969)
Dark Ages:
Self-organization in visual cortex (von der Malsburg; 1973)
Backpropagation (Werbos, 1974)
Foundations of Adaptive Resonance Theory (Grossberg; 1976)
Neural Theory of Association (Amari; 1977)

6. History Modern Ages: Adaptive Resonance Theory (Grossberg; 1980)
Hopfield model (Hopfield; 1982, 1984)
Self-organizing maps (Kohonen; 1982)
Reinforcement learning (Sutton and Barto; 1983)
Simulated Annealing (Kirkpatrick et al.; 1983)
Boltzmann machines (Ackley, Hinton, Terrence; 1985)
Backpropagation (Rumelhart, Hinton, Williams; 1986)
ART-networks (Carpenter, Grossberg; 1992)
Support Vector Machines

7. Hebb’s Learning Law
In 1949, Donald Hebb formulated William James’ principle of association into a mathematical form.
If the activation of the neurons, y1 and y2 , are both on (+1) then the weight between the two neurons grow. (Off: 0)
Else the weight between remains the same.
However, when bipolar activation {-1,+1} scheme is used, then the weights can also decrease when the activation of two neurons does not match.

8. Real Neural Learning
Synapses change size and strength with experience.
Hebbian learning: When two connected neurons are firing at the same time, the strength of the synapse between them increases.
“Neurons that fire together, wire together.”

9. Biological Neurons Human brain = tens of thousands of neurons
Each neuron is connected to thousands other neurons
A neuron is made of:
The soma: body of the neuron
Dendrites: filaments that provide input to the neuron
The axon: sends an output signal
Synapses: connection with other neurons – releases certain quantities of chemicals called neurotransmitters to other neurons

10. Modeling of Brain Functions

11. The biological neuron
The pulses generated by the neuron travels along the axon as an electrical wave.
Once these pulses reach the synapses at the end of the axon open up chemical vesicles exciting the other neuron.

12. How do NNs and ANNs work?
Information is transmitted as a series of electric impulses, so-called spikes.
The frequency and phase of these spikes encodes the information.
In biological systems, one neuron can be connected to as many as 10,000 other neurons.
Usually, a neuron receives its information from other neurons in a confined area

13. Navigation of a car
Done by Pomerlau. The network takes inputs from a 34X36 video image and a 7X36 range finder. Output units represent “drive straight”, “turn left” or “turn right”. After training about 40 times on 1200 road images, the car drove around CMU campus at 5 km/h (using a small workstation on the car). This was almost twice the speed of any other non-NN algorithm at the time.

14. Automated driving at 70 mph on a public highway
Camera
image
30 outputs
for steering
30x32 weights
into one out of
four hidden
unit
4 hidden
units
30x32 pixels
as inputs

15. Computers vs. Neural Networks
“Standard” Computers Neural Networks
one CPU highly parallel processing
fast processing units slow processing units
reliable units unreliable units
static infrastructure dynamic infrastructure

16. Neural Network

17. Neural Network Application
Pattern recognition can be implemented using NN
The figure can be T or H character, the network should identify each class of T or H.

21. Simple Neuron X1
Inputs X2
Output Xn b

22. An Artificial Neuron x1 synapses neuron i x2 Wi,1 Wi,2 … xi … Wi,n xn
net input signal
output

23. Neural Network
Input Layer
Hidden 1
Hidden 2
Output Layer

24. Network Layers
The common type of ANN consists of three layers of neurons: a layer of input neurons connected to the layer of hidden neuron which is connected to a layer of output neurons.

25. Architecture of ANN Feed-Forward networks
Allow the signals to travel one way from input to output
Feed-Back Networks
The signals travel as loops in the network, the output is connected to the input of the network

29. How do NNs and ANNs Learn?
NNs are able to learn by adapting their connectivity patterns so that the organism improves its behavior in terms of reaching certain (evolutionary) goals.
The NN achieves learning by appropriately adapting the states of its synapses.

30. Learning Rule
The learning rule modifies the weights of the connections.
The learning process is divided into Supervised and Unsupervised learning

31. Supervised Network
Which means there exists an external teacher. The target is to minimization of the error between the desired and computed output

35. Unsupervised Network
Uses no external teacher and is based upon only local information.

39. Perceptron
It is a network of one neuron and hard limit transfer function
Inputs
 f X1 X2 Xn
Output W1 W2 Wn

40. Perceptron The perceptron is given first a randomly weights vectors
Perceptron is given chosen data pairs (input and desired output)
Preceptron learning rule changes the weights according to the error in output

42. Perceptron Learning Rule
W new = W old + (t-a) X
Where W new is the new weight
W old is the old value of weight
X is the input value
t is the desired value of output
a is the actual value of output

43. Example Let W = [2 2] and b = -3 X1 = [0 0] and t =0

44. AND Network
This example means we construct a network for AND operation. The network draw a line to separate the classes which is called Classification

45. Perceptron Geometric View
The equation below describes a (hyper-)plane in the input space consisting of real valued m-dimensional vectors. The plane splits the input space into two regions, each of them describing one class.
decision
region for C1 x2
w1x1 + w2x2 + w0 >= 0
decision
boundary C1 x1 C2
w1x1 + w2x2 + w0 = 0

46. Problems Four one-dimensional data belonging to two classes are
X = [ ]
T = [ ]
W = [ ]

47. Boolean Functions Take in two inputs (-1 or +1)
Produce one output (-1 or +1)
In other contexts, use 0 and 1
Example: AND function
Produces +1 only if both inputs are +1
Example: OR function
Produces +1 if either inputs are +1
Related to the logical connectives from F.O.L.

48. The First Neural Neural Networks
AND Function 1 X1 X2 Y
Threshold(Y) = 2

49. Simple Networks
t = 0.0 y x
W = 1.5
W = 1 -1

50. Exercises Design a neural network to recognize the problem of
X1=[2 2] , t1=0
X=[1 -2], t2=1
X3=[-2 2], t3=0
X4=[-1 1], t4=1
Start with initial weights w=[0 0] and bias =0

51. Perceptron: Limitations
The perceptron can only model linearly separable classes, like (those described by) the following Boolean functions:
AND OR
COMPLEMENT
It cannot model the XOR.
You can experiment with these functions in the Matlab practical lessons.

52. Types of decision regionsx1 1 x2 w2 w1 w0
Network
with a single
node 1 x1 x2
Convex
region L1 L2 L3 L4
One-hidden layer network that realizes the convex region
-3.5

53. Gaussian Neurons
Another type of neurons overcomes this problem by using a Gaussian activation function: 1
fi(neti(t))
neti(t) -1

54. 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.

55. 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

56. Sigmoidal Neurons fi(neti(t))  = 1 neti(t)
fi(neti(t))
neti(t) -1
 = 1
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.

57. Sigmoidal Neurons
This leads to a simplified form of the sigmoid function:
We do not need a modifiable threshold , because we will use “dummy” inputs as we did for perceptrons.
The choice  = 1 works well in most situations and results in a very simple derivative of S(net).

58. Sigmoidal Neurons
This result will be very useful when we develop the backpropagation algorithm.

59. Multi-layers Network Let the network of 3 layers
Input layer
Hidden layer
Output layer
Each layer has different number of neurons
The famous example to need the multi-layer network is XOR unction

64. Learning rule
The perceptron learning rule can not be applied to multi-layer network
We use BackPropagation Algorithm in learning process

65. Feed-forward + Backpropagation
input from the features is fed forward in the network from input layer towards the output layer
Backpropagation:
Method to asses the blame of errors to weights
error rate flows backwards from the output layer to the input layer (to adjust the weight in order to minimize the output error)

66. Backprop Back-propagation training algorithm illustrated:
Backprop adjusts the weights of the NN in order to minimize the network total mean squared error.
Network activation
Error computation
Forward Step
Error propagation
Backward Step

67. Correlation Learning Hebbian Learning (1949):
“When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes place in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.”
Weight modification rule:
wi,j = cxixj
Eventually, the connection strength will reflect the correlation between the neurons’ outputs.

68. Competitive Learning Nodes compete for inputs
Node with highest activation is the winner
Winner neuron adapts its tuning (pattern of weights) even further towards the current input
Individual nodes specialize to win competition for a set of similar inputs
Process leads to most efficient neural representation of input space
Typical for unsupervised learning

69. Backpropagation Learning
Similar to the Adaline, the goal of the Backpropagation learning algorithm is to modify the network’s weights so that its output vector
op = (op,1, op,2, …, op,K)
is as close as possible to the desired output vector
dp = (dp,1, dp,2, …, dp,K)
for K output neurons and input patterns p = 1, …, P.
The set of input-output pairs (exemplars) {(xp, dp) | p = 1, …, P} constitutes the training set.

70. Bp Algorithm The weight change rule is
Where  is the learning factor <1
Error is the error between actual and trained value
f’ is is the derivative of sigmoid function = f(1-f)

71. Delta Rule
Each observation contributes a variable amount to the output
The scale of the contribution depends on the input
Output errors can be blamed on the weights
A least mean square (LSM) error function can be defined (ideally it should be zero)
E = ½ (t – y)2

72. Example
For the network with one neuron in input layer and one neuron in hidden layer the following values are given
X=1, w1 =1, b1=-2, w2=1, b2 =1, =1 and t=1
Where X is the input value
W1 is the weight connect input to hidden
W2 is the weight connect hidden to output
B1 and b2 are bias
T is the training value

73. Exercises Design a neural network to recognize the problem of
X1=[2 2] , t1=0
X=[1 -2], t2=1
X3=[-2 2], t3=0
X4=[-1 1], t4=1
Start with initial weights w=[0 0] and bias =0

74. Exercises
Perform one iteration of backprpgation to network of two layers. First layer has one neuron with weight 1 and bias –2. The transfer function in first layer is f=n2
The second layer has only one neuron with weight 1 and bias 1. The f in second layer is 1/n.
The input to the network is x=1 and t=1

75. Neural Network Construct a neural network to solve the problem X1 X2
Output
1.0 1
9.4
6.4 -1
2.5
2.1
8.0
7.7
0.5
2.2
7.9
8.4
7.0
2.8
0.8
1.2
3.0
7.8
6.1
Initialize the weights 0.75 , 0.5, and –0.6

76. Neural Network Construct a neural network to solve the XOR problem X1
Output 1
Initialize the weights –7.0 , -7.0, -5.0 and –4.0

77. -0.5
The transfer function is linear function. -2 1 1 1 -1 -1 1 3
0.5
-0.5

78. Consider a transfer function as f(n) = n2
Consider a transfer function as f(n) = n2. Perform one iteration of BackPropagation with a= 0.9 for neural network of two neurons in input layer and one neuron in output layer. The input values are X=[1 -1] and t = 8, the weight values between input and hidden layer are w11 = 1, w12 = - 2, w21 = 0.2, and w22 = 0.1. The weight between input and output layers are w1 = 2 and w2= -2. The bias in input layers are b1 = -1, and b2= 3.

79. Some variations
True gradient descent assumes infinitesmall learning rate (). If  is too small then learning is very slow. If large, then the system's learning may never converge.
Some of the possible solutions to this problem are:
Add a momentum term to allow a large learning rate.
Use a different activation function
Use a different error function
Use an adaptive learning rate
Use a good weight initialization procedure.
Use a different minimization procedure

80. Problems with Local Minima
Backpropagation is gradient descent search
Where the height of the hills is determined by error
But there are many dimensions to the space
One for each weight in the network
Therefore backpropagation
Can find its ways into local minima
One partial solution:
Random re-start: learn lots of networks
Starting with different random weight settings
Can take best network
Or can set up a “committee” of networks to categorise examples
Another partial solution: Momentum

81. Adding Momentum Imagine rolling a ball down a hill
Without Momentum With Momentum
Gets stuck
here

82. Momentum in Backpropagation
For each weight
Remember what was added in the previous epoch
In the current epoch
Add on a small amount of the previous Δ
The amount is determined by
The momentum parameter, denoted α
α is taken to be between 0 and 1

83. How Momentum Works If direction of the weight doesn’t change
Then the movement of search gets bigger
The amount of additional extra is compounded in each epoch
May mean that narrow local minima are avoided
May also mean that the convergence rate speeds up
Caution:
May not have enough momentum to get out of local minima
Also, too much momentum might carry search
Back out of the global minimum, into a local minimum

84. Weight update becomes:  wij (n+1) =  (pj opi) +   wij(n)
Momentum
Weight update becomes:
 wij (n+1) =  (pj opi) +   wij(n)
The momentum parameter  is chosen between 0 and 1, typically 0.9. This allows one to use higher learning rates. The momentum term filters out high frequency oscillations on the error surface.
What would the learning rate be in a deep valley?

85. Problems with Overfitting
Plot training example error versus test example error:
Test set error is increasing!
Network is overfitting the data
Learning idiosyncrasies in data, not general principles
Big problem in Machine Learning (ANNs in particular)

86. Avoiding Overfitting
Bad idea to use training set accuracy to terminate
One alternative: Use a validation set
Hold back some of the training set during training
Like a miniature test set (not used to train weights at all)
If the validation set error stops decreasing, but the training set error continues decreasing
Then it’s likely that overfitting has started to occur, so stop
Another alternative: use a weight decay factor
Take a small amount off every weight after each epoch
Networks with smaller weights aren’t as highly fine tuned (overfit)