Introduction to Neural
Networks
John Paxton
Montana State University
Summer 2003
Chapter 2: Simple Neural Networks
for Pattern Classification
1
x0
w0
x1
w1
w0 is the bias
y
f(yin) = 1 if yin >= 0
f(yin) = 0 otherwise
ARCHITECTURE
xn
wn
Representations
• Binary:
0 no, 1 yes
• Bipolar: -1 no, 0 unknown, 1 yes
• Bipolar is superior
Interpreting the Weights
• w0 = -1, w1 = 1, w2 = 1
• 0 = -1 + x1 + x2 or x2 = 1 – x1
YES
x1
NO
x2
decision boundary
Modelling a Simple Problem
• Should I attend this lecture?
• x1 = it’s hot
x
• x2 = it’s raining
0
x1
x2
2.5
-2
1
y
Linear Separability
0
1
0
0
AND
1
1
0
1
OR
1
0
0
1
XOR
Hebb’s Rule
• 1949. Increase the weight between two
neurons that are both “on”.
• 1988. Increase the weight between two
neurons that are both “off”.
• wi(new) = wi(old) + xi*y
Algorithm
1. set wi = 0 for 0 <= i <= n
2. for each training vector
3. set xi = si for all input units
4. set y = t
5. wi(new) = wi(old) + xi*y
Example: 2 input AND
s0
s1
s2
t
1
1
1
1
1
1
-1
-1
1
-1
1
-1
1
-1
-1
-1
Training Procedure
w0
w1
w2
x0
x1
x2
y
0
0
0
1
1
1
1
1
1
1
1
1
-1
-1 (!)
0
0
2
1
-1
1
-1 (!)
-1
1
1
1
-1
-1
-1
-2
2
2
Result Interpretation
• -2 + 2x1 + 2x2 = 0 OR
• x2 = -x1 + 1
• This training procedure is order dependent
and not guaranteed.
Pattern Recognition Exercise
• #.#
.#.
#.#
“X”
.#.
#.#
.#.
“O”
Pattern Recognition Exercise
• Architecture?
• Weights?
• Are the original patterns classified
correctly?
• Are the original patterns with 1 piece of
wrong data classified correctly?
• Are the original patterns with 1 piece of
missing data classified correctly?
Perceptrons (1958)
• Very important early neural network
• Guaranteed training procedure under
certain circumstances
1
x0
w0
x1
xn
w1
wn
y
Activation Function
• f(yin) = 1 if yin > q
f(yin) = 0 if -q <= yin <= q
f(yin) = -1 otherwise
• Graph interpretation
1
-1
Learning Rule
• wi(new) = wi(old) + a*t*xi
• a is the learning rate
• Typically, 0 < a <= 1
if error
Algorithm
1. set wi = 0 for 0 <= i <= n (can be random)
2. for each training exemplar do
3. xi = si
4. yin = S xi*wi
5. y = f(yin)
6. wi(new) = wi(old) + a*t*xi if error
7. if stopping condition not reached, go to 2
Example: AND concept
•
•
•
•
bipolar inputs
bipolar target
q=0
a=1
Epoch 1
w0
w1
w2
x0
x1
x2
y
t
0
0
0
1
1
1
0
1
1
1
1
1
1
-1
1
-1
0
0
2
1
-1
1
1
-1
-1
1
1
1
-1
-1
-1
-1
Exercise
• Continue the above example until the
learning algorithm is finished.
Perceptron Learning Rule
Convergence Theorem
• If a weight vector exists that correctly
classifies all of the training examples, then
the perceptron learning rule will converge
to some weight vector that gives the
correct response for all training patterns.
This will happen in a finite number of
steps.
Exercise
• Show perceptron
weights for the 2-of-3
concept
x1
1
1
1
1
-1
-1
-1
-1
x2
1
1
-1
-1
1
1
-1
-1
x3
1
-1
1
-1
1
-1
1
-1
y
1
1
1
-1
1
-1
-1
-1
Adaline (Widrow, Huff 1960)
• Adaptive Linear Network
• Learning rule minimizes the mean squared
error
• Learns on all examples, not just ones with
errors
Architecture
1
x0
w0
x1
xn
w1
wn
y
Training Algorithm
1. set wi (small random values typical)
2. set a (0.1 typical)
3. for each training exemplar do
4. xi = si
5. yin = S xi*wi
6. wi(new) = wi(old) + a*(t – yin)*xi
7. go to 3 if largest weight change big
enough
Activation Function
• f(yin) = 1 if yin >= 0
• f(yin) = -1 otherwise
Delta Rule
• squared error E = (t – yin)2
• minimize error E’ = -2(t – yin)xi
= a(t – yin)xi
Example: AND concept
• bipolar inputs
• bipolar targets
• w0 = -0.5, w1 = 0.5,
w2 = 0.5
• minimizes E
x0
x1
x2
yin
t
E
1
1
1
.5
1 .25
1
1
-1
-.5
-1 .25
1
-1
1
-.5
-1 .25
1
-1
-1
-1.5 -1 .25
Exercise
• Demonstrate that you understand the
Adaline training procedure.
Madaline
• Many adaptive linear neurons
1
1
y
x1
z1
xm
zk
Madaline
• MRI (1960) – only learns weights from
input layer to hidden layer
• MRII (1987) – learns all weights