1. Artificial Neural Networks
Notes based on Nilsson and Mitchell’s
Machine learning

2. Outline Perceptrons (LTU) Gradient descent Multi-layer networks
Backpropagation

3. Biological Neural Systems
Neuron switching time : > 10-3 secs
Number of neurons in the human brain: ~1010
Connections (synapses) per neuron : ~104–105
Face recognition : 0.1 secs
High degree of parallel computation
Distributed representations

4. Properties of Artificial Neural Nets (ANNs)
Many simple neuron-like threshold switching units
Many weighted interconnections among units
Highly parallel, distributed processing
Learning by adaptation of the connection weights

5. Appropriate Problem Domains for Neural Network Learning
Input is high-dimensional discrete or real-valued (e.g. raw sensor input)
Output is discrete or real valued
Output is a vector of values
Form of target function is unknown
Humans do not need to interpret the results (black box model)

6. General Idea A network of neurons. Each neuron is characterized by:
number of input/output wires
weights on each wire
threshold value
These values are not explicitly programmed, but they evolve through a training process.
During training phase, labeled samples are presented. If the network classifies correctly, no weight changes. Otherwise, the weights are adjusted.
backpropagation algorithm used to adjust weights.

7. ALVINN (Carnegie Mellon Univ)
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

8. Another Example
NETtalk – Program that learns to pronounce English text. (Sejnowski and Rosenberg 1987).
A difficult task using conventional programming models.
Rule-based approaches are too complex since pronunciations are very irregular.
NETtalk takes as input a sentence and produces a sequence of phonemes and an associated stress for each letter.

9. NETtalk
A phoneme is a basic unit of sound in a language.
Stress – relative loudness of that sound.
Because the pronunciation of a single letter depends upon its context and the letters around it, NETtalk is given a seven character window.
Each position is encoded by one of 29 symbols, (26 letters and 3 punctuations.)
Letter in each position activates the corresponding unit.

10. NETtalk
The output units encode phonemes using 21 different features of human articulation.
Remaining five units encode stress and syllable boundaries.
NETtalk also has a middle layer (hidden layer) that has 80 hidden units and nearly connections (edges).
NETtalk is trained by giving it a 7 character window so that it learns the pronounce the middle character.
It learns by comparing the computed pronunciation to the correct pronunciation.

11. Handwritten character recognition
This is another area in which neural networks have been successful.
In fact, all the successful programs have a neural network component.

12. Threshold Logic Unit (TLU)
inputs
weights x1 w1
output
activation w2 x2  y . q
a=i=1n wi xi wn xn
1 if a  q y=
0 if a < q {

14. Activation Functions threshold linear y y a a sigmoid y
piece-wise linear y a a

15. Decision Surface of a TLU1 1
Decision line
w1 x1 + w2 x2 = q x2 w 1 x1 1

16. Scalar Products & Projectionsv v v w j w j w j
w • v > 0
w • v = 0
w • v < 0 v w j
w • v = |w||v| cos j

17. Geometric Interpretation
The relation w•x=q defines the decision line x2
Decision line w
y=1
w•x=q
|xw|=q/|w| xw x1 x
y=0

18. Geometric Interpretation
In n dimensions the relation w • x=q defines a n-1 dimensional hyper-plane, which is perpendicular to the weight vector w.
On one side of the hyper-plane (w • x > q) all patterns are classified by the TLU as “1”, while those that get classified as “0” lie on the other side of the hyper-plane.
If patterns can be not separated by a hyper-plane then they cannot be correctly classified with a TLU.

19. Linear Separability x2 w1=? w2=? q= ? w1=1 w2=1 q=1.5 1 1 x1 x1 11 1 x1 x1 1
Logical XOR
Logical AND x1 x2 a y 1 2 x1 x2 y 1

20. {  . Threshold as Weight q=wn+1 a= i=1n+1 wi xi xn+1=-1 x1 w1 wn+1y .
a= i=1n+1 wi xi wn xn
1 if a  0
y =
0 if a <0 {

21. Geometric Interpretation
The relation w • x=0 defines the decision line x2
Decision line w
w•x=0
y=1 x1
y=0 x

22. Training ANNs Training set S of examples {x, t}
x is an input vector and
t the desired target vector
Example: Logical And
S = {(0,0),0}, {(0,1),0}, {(1,0),0}, {(1,1),1}
Iterative process
Present a training example x , compute network output y, compare output y with target t, adjust weights and thresholds
Learning rule
Specifies how to change the weights w and thresholds q of the network as a function of the inputs x, output y and target t.

23. Adjusting the Weight Vectorx x
w’ = w + ax
j>90 ax w w
Target t=1
Output y=0
Move w in the direction of x x x w
-ax
j<90 w
w’ = w - ax
Target t=0
Output y=1
Move w away from the direction of x

24. Perceptron Learning Rule
w’= w + a (t-y) x
Or in components
w’i = wi + Dwi = wi + a (t-y) xi (i=1..n+1)
With wn+1 = q and xn+1= –1
The parameter a is called the learning rate. It determines the magnitude of weight updates Dwi .
If the output is correct (t = y) the weights are not changed (Dwi =0).
If the output is incorrect (t  y) the weights wi are changed such that the output of the TLU for the new weights w’i is closer/further to the input xi.

25. Perceptron Training Algorithm
repeat
for each training vector pair (x,t)
evaluate the output y when x is the input
if yt then
form a new weight vector w’ according
to w’=w + a (t-y) x
else
do nothing
end if
end for
until y=t for all training vector pairs

26. Perceptron Convergence Theorem
The algorithm converges to the correct classification
if the training data is linearly separable
and a is sufficiently small
If two classes of vectors X1 and X2 are linearly separable, the application of the perceptron training algorithm will eventually result in a weight vector w0, such that w0 defines a TLU whose decision hyper-plane separates X1 and X2 (Rosenblatt 1962).
Solution w0 is not unique, since if w0 x =0 defines a hyper-plane, so does w’0 = k w0.

27. Example x1 x2 output Initial weights: (0.75, -0.5, -0.6) 1 1
Initial weights: (0.75, -0.5, -0.6)

28. . . . . . . Multiple TLUs x1 x2 x3 xn w’ji = wji + a (tj-yj) xi
Handwritten alphabetic character recognition
26 classes : A,B,C…,Z
First TLU distinguishes between “A”s and “non-A”s, second TLU between “B”s and “non-B”s etc.
. . . y1 y2
y26
wji connects xi with yj
. . . x1 x2 x3 xn
w’ji = wji + a (tj-yj) xi

29.  . Linear Unit x1 w1 w2 x2 y y= a = i=1n wi xi a=i=1n wi xi wn xn
inputs
weights x1 w1
output
activation w2 x2  y .
y= a = i=1n wi xi
a=i=1n wi xi wn xn

30. Gradient Descent Learning Rule
Consider linear unit without threshold and continuous output o (not just –1,1)
0 =w0 + w1 x1 + … + wn xn
Train the wi’s such that they minimize the squared error
e = aD (fa-da)2
where D is the set of training examples
Here fa is the actual output, da is the desired output.

31. Gradient Descent rule:
We want to choose the weights wi so that e is minimized. Recall that
e = aD (fa – da)2
Since our goal is to work this error function for one input at a time, let us consider a fixed input x in D, and define
e = (fx – dx)2
We will drop the subscript and just write this as:
e = (f – d)2
Our goal is to find the weights that will minimize this expression.

32. e/W= [e/w1,… e/wn+1]
Since s, the threshold function, is given by s = X . W, we have:
e/W = e/s * s/W. However, s/W = X. Thus,
e/W = e/s * X
Recall from the previous slide that e = (f – d)2
So, we have: e/s = 2(f – d)* f /s (note: d is constant)
This gives the expression:
e/W = 2(f – d)* f /s * X
A problem arises when dealing with TLU, namely f is not a continuous function of s.

33. For a fixed input x, suppose the desired output is d, and the actual output is f, then the above expression becomes:
w = - 2(d – f) x
This is what is known as the Widrow-Hoff procedure, with 2 replaced by c:
The key idea is to move the weight vector along the gradient.
When will this converge to the correct weights?
We are assuming that the data is linearly separable.
We are also assuming that the desired output from the linear threshold gate is available for the training set.
Under these conditions, perceptron convergence theorem shows that the above procedure will converge to the correct weights after a finite number of iterations.

34. Neuron with Sigmoid-Function
inputs
weights x1 w1
output
activation w2 x2  y .
a=i=1n wi xi wn xn
y=s(a) =1/(1+e-a)

35. Sigmoid Unit  . x0=-1 x1 w1 w0 w2 x2 y wn xn a=i=0n wi xi
y=(a)=1/(1+e-a) w2 x2  y .
(x) is the sigmoid function: 1/(1+e-x) wn
d(x)/dx= (x) (1- (x)) xn
Derive gradient descent rules to train:

36. Sigmoid function f s

37. Gradient Descent Rule for Sigmoid Output Function
Ep[w1,…,wn] = (tp-yp)2
Ep/wi = /wi (tp-yp)2
= /wi(tp- s(Si wi xip))2
= (tp-yp) s’(Si wi xip) (-xip)
for y=s(a) = 1/(1+e-a)
s’(a)= e-a/(1+e-a)2=s(a) (1-s(a)) a s’ a
w’i= wi + wi = wi + a y(1-y)(tp-yp) xip

38. Presentation of Training Examples
Presenting all training examples once to the ANN is called an epoch.
In incremental stochastic gradient descent training examples can be presented in
Fixed order (1,2,3…,M)
Randomly permutated order (5,2,7,…,3)
Completely random (4,1,7,1,5,4,……) (repetitions allowed arbitrarily)

39. Capabilities of Threshold Neurons
The threshold neuron can realize any linearly separable function Rn  {0, 1}.
Although we only looked at two-dimensional input, our findings apply to any dimensionality n.
For example, for n = 3, our neuron can realize any function that divides the three-dimensional input space along a two-dimension plane.

40. Capabilities of Threshold Neurons
What do we do if we need a more complex function?
We can combine multiple artificial neurons to form networks with increased capabilities.
For example, we can build a two-layer network with any number of neurons in the first layer giving input to a single neuron in the second layer.
The neuron in the second layer could, for example, implement an AND function.

41. Capabilities of Threshold Neurons. oi
What kind of function can such a network realize?

42. Capabilities of Threshold Neurons
Assume that the dotted lines in the diagram represent the input-dividing lines implemented by the neurons in the first layer:
1st comp.
2nd comp.
Then, for example, the second-layer neuron could output 1 if the input is within a polygon, and 0 otherwise.

43. Capabilities of Threshold Neurons
However, we still may want to implement functions that are more complex than that.
An obvious idea is to extend our network even further.
Let us build a network that has three layers, with arbitrary numbers of neurons in the first and second layers and one neuron in the third layer.
The first and second layers are completely connected, that is, each neuron in the first layer sends its output to every neuron in the second layer.

44. Capabilities of Threshold Neurons. oi
What type of function can a three-layer network realize?

45. Capabilities of Threshold Neurons
Assume that the polygons in the diagram indicate the input regions for which each of the second-layer neurons yields output 1:
1st comp.
2nd comp.
Then, for example, the third-layer neuron could output 1 if the input is within any of the polygons, and 0 otherwise.

46. Capabilities of Threshold Neurons
The more neurons there are in the first layer, the more vertices can the polygons have.
With a sufficient number of first-layer neurons, the polygons can approximate any given shape.
The more neurons there are in the second layer, the more of these polygons can be combined to form the output function of the network.
With a sufficient number of neurons and appropriate weight vectors wi, a three-layer network of threshold neurons can realize any function Rn  {0, 1}.

47. Terminology
Usually, we draw neural networks in such a way that the input enters at the bottom and the output is generated at the top.
Arrows indicate the direction of data flow.
The first layer, termed input layer, just contains the input vector and does not perform any computations.
The second layer, termed hidden layer, receives input from the input layer and sends its output to the output layer.
After applying their activation function, the neurons in the output layer contain the output vector.

48. Terminology Example: Network function f: R3  {0, 1}2 output vector
output layer
hidden layer
input layer
input vector

49. Multi-Layer Networks
output layer
hidden layer
input layer

50. Training-Rule for Weights to the Output Layer
Ep[wij] = ½ j (tjp-yjp)2 yj
Ep/wij = /wij ½ Sj (tjp-yjp)2
= …
= - yjp(1-ypj)(tpj-ypj) xip
wji xi
wij = a yjp(1-yjp) (tpj-yjp) xip
= a djp xip
with djp := yjp(1-yjp) (tpj-yjp)

51. Training-Rule for Weights to the Hidden Layeryj
Credit assignment problem:
No target values t for hidden layer units. dj
wjk xk
Error for hidden units? dk
dk = Sj wjk dj yj (1-yj)
wki
wki = a xkp(1-xkp) dkp xip xi

52. Training-Rule for Weights to the Hidden Layeryj
Ep[wki] = ½ j (tjp-yjp)2 dj
wjk
Ep/wki = /wki ½ Sj (tjp-yjp)2
=/wki ½Sj (tjp-s(Skwjk xkp))2
=/wki ½Sj (tjp-s(Skwjk s(Siwki xip)))2
= -j (tjp-yjp) s’j(a) wjk s’k(a) xip
= -j dj wjk s’k(a) xip
= -j dj wjk xk (1-xk) xip xk dk
wki xi
wki = a dk xip
with dk = j dj wjk xk(1-xk)

53. Backpropagation Backward step:
propagate errors from output to hidden layer yj dj
wjk xk dk
wki
Forward step:
Propagate activation
from input to output layer xi

54. Backpropagation Algorithm
Initialize weights wij with a small random value
repeat
for each training pair {(x1,…xn)p,(t1,...,tm)p} Do
Present (x1,…,xn)p to the network and compute the outputs yj (forward step)
Compute the errors dj in the output layer and propagate them to the hidden layer (backward step)
Update the weights in both layers according to
wki = a dk xi
end for loop
until overall error E becomes acceptably low

55. Backpropagation Algorithm
Initialize each wi to some small random value
Until the termination condition is met, Do
For each training example <(x1,…xn),t> Do
Input the instance (x1,…,xn) to the network and compute the network outputs yk
For each output unit k
k=yk(1-yk)(tk-yk)
For each hidden unit h
h=yh(1-yh) k wh,k k
For each network weight wi,j Do
wi,j=wi,j+wi,j where
wi,j=  j xi,j

56. Backpropagation Gradient descent over entire network weight vector
Easily generalized to arbitrary directed graphs
Will find a local, not necessarily global error minimum
-in practice often works well (can be invoked multiple times with different initial weights)
Often include weight momentum term
wi,j(n)=  j xi,j +  wi,j (n-1)
Minimizes error training examples
Will it generalize well to unseen instances (over-fitting)?
Training can be slow typical iterations
(Using network after training is fast)

57. Backpropagation
Easily generalized to arbitrary directed graphs without clear layers.
BP finds a local, not necessarily global error minimum
- in practice often works well (can be invoked multiple times with different initial weights)
Minimizes error over training examples
How does it generalize to unseen instances ?
Training can be slow typical iterations
(use more efficient optimization methods than gradient descent)
Using network after training is fast

58. Convergence of Backprop
Gradient descent to some local minimum perhaps not global minimum
Add momentum term: wki(n)
wki(n) = a dk(n) xi (n) + l Dwki(n-1)
with l  [0,1]
Stochastic gradient descent
Train multiple nets with different initial weights
Nature of convergence
Initialize weights near zero
Therefore, initial networks near-linear
Increasingly non-linear functions possible as training progresses

59. Expressive Capabilities of ANN
Boolean functions
Every boolean function can be represented by network with single hidden layer
But might require exponential (in number of inputs) hidden units
Continuous functions
Every bounded continuous function can be approximated with arbitrarily small error, by network with one hidden layer [Cybenko 1989, Hornik 1989]
Any function can be approximated to arbitrary accuracy by a network with two hidden layers [Cybenko 1988]