Unsupervised Learning
So far, we have only looked at supervised learning, in
which an external teacher improves network
performance by comparing desired and actual outputs
and modifying the synaptic weights accordingly.
However, most of the learning that takes place in our
brains is completely unsupervised.
This type of learning is aimed at achieving the most
efficient representation of the input space, regardless
of any output space.
Unsupervised learning can also be useful in artificial
neural networks.
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
1
Unsupervised Learning
Applications of unsupervised learning include
• Clustering
• Vector quantization
• Data compression
• Feature extraction
Unsupervised learning methods can also be combined
with supervised ones to enable learning through inputoutput pairs like in the BPN.
One such hybrid approach is the counterpropagation
network.
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
2
Unsupervised/Supervised Learning:
The Counterpropagation Network
The counterpropagation network (CPN) is a fastlearning combination of unsupervised and supervised
learning.
Although this network uses linear neurons, it can
learn nonlinear functions by means of a hidden layer
of competitive units.
Moreover, the network is able to learn a function and
its inverse at the same time.
However, to simplify things, we will only consider the
feedforward mechanism of the CPN.
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
3
Distance/Similarity Functions
In the hidden layer, the neuron whose weight vector is
most similar to the current input vector is the “winner.”
There are different ways of defining such maximal
similarity, for example:
(1) Maximal cosine similarity (same as net input):
s(w, x)  w  x
(2) Minimal Euclidean distance:
d (w, x)   wi  xi 
2
i
(no square root necessary for determining the winner)
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
4
The Counterpropagation Network
A simple CPN with two input neurons, three hidden neurons,
and two output neurons can be described as follows:
O
w21
O
13
Y1 w
O
11
w
O
w12
H1
H
12
w
Y2
w
O
w22
H
w21
H2
H
w31
H3
H
w22
H
w32
H
11
w
X1
November 9, 2010
Output
layer
O
23
X2
Neural Networks
Lecture 16: Counterpropagation
Hidden
layer
Input
layer
5
The Counterpropagation Network
The CPN learning process (general form for n
input units and m output units):
1. Randomly select a vector pair (x, y) from the
training set.
2. If you use the cosine similarity function, normalize
(shrink/expand to “length” 1) the input vector x by
dividing every component of x by the magnitude
||x||, where
|| x || 
n
2
x
 j
j 1
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
6
The Counterpropagation Network
3. Initialize the input neurons with the resulting vector
and compute the activation of the hidden-layer
units according to the chosen similarity measure.
4. In the hidden (competitive) layer, determine the unit
W with the largest activation (the winner).
5. Adjust the connection weights between W and all N
input-layer units according to the formula:
w (t  1)  w (t )   ( xn  w (t ))
H
Wn
H
Wn
H
Wn
6. Repeat steps 1 to 5 until all training patterns have
been processed once.
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
7
The Counterpropagation Network
7. Repeat step 6 until each input pattern is
consistently associated with the same competitive
unit.
8. Select the first vector pair in the training set (the
current pattern).
9. Repeat steps 2 to 4 (normalization, competition)
for the current pattern.
10. Adjust the connection weights between the
winning hidden-layer unit and all M output layer
units according to the equation:
O
O
O
wmW
(t  1)  wmW
(t )   ( ym  wmW
(t ))
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
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The Counterpropagation Network
11. Repeat steps 9 and 10 for each vector pair in the
training set.
12. Repeat steps 8 through 11 for several epochs.
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
9
The Counterpropagation Network
Because in our example network the input is twodimensional, each unit in the hidden layer has two
weights (one for each input connection).
Therefore, input to the network as well as weights of
hidden-layer units can be represented and visualized
by two-dimensional vectors.
For the current network, all weights in the hidden layer
can be completely described by three 2D vectors.
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
10
Counterpropagation – Cosine Similarity
This diagram shows a sample state of the hidden layer and a
sample input to the network:
H
H
( w21
, w22
)
( w11H , w12H )
( x1 , x2 )
H
H
( w31
, w32
)
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
11
Counterpropagation – Cosine Similarity
In this example, hidden-layer neuron H2 wins and, according to
the learning rule, is moved closer towards the current input
vector.
H
H
( w21
, w22
)
( w11H , w12H )
H
H
(w21
,
w
, new
22, new )
( x1 , x2 )
H
H
( w31
, w32
)
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
12
Counterpropagation – Cosine Similarity
After doing this through many epochs and slowly reducing the
adaptation step size , each hidden-layer unit will win for a
subset of inputs, and the angle of its weight vector will be in the
center of gravity of the angles of these inputs.
all input vectors
in the training set
H
H
( w21
, w22
)
( w11H , w12H )
H
H
( w31
, w32
)
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
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Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ +
+
3
x
1
x
November 9, 2010
x
x
Neural Networks
Lecture 16: Counterpropagation
2
o
o o
o
14
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ +
+
3
x
1
x
November 9, 2010
x
x
Neural Networks
Lecture 16: Counterpropagation
2
o
o o
o
15
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ +
+
3
x
x
November 9, 2010
1
x
x
Neural Networks
Lecture 16: Counterpropagation
2
o
o o
o
16
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ +
+
3
x
x
November 9, 2010
1
x
x
Neural Networks
Lecture 16: Counterpropagation
2
o
o o
o
17
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ +
+
3
x
x
November 9, 2010
1
x
x
Neural Networks
Lecture 16: Counterpropagation
2
o
o o
o
18
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ +
+
3
x
x
November 9, 2010
1
x
x
Neural Networks
Lecture 16: Counterpropagation
2
o
o o
o
19
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ +
+
1
3
x
x
November 9, 2010
x
x
Neural Networks
Lecture 16: Counterpropagation
2
o
o o
o
20
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ +
+
1
3
x
x
November 9, 2010
x
x
Neural Networks
Lecture 16: Counterpropagation
2
o
o o
o
21
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ 1+
+
o
o o
o
3
x
x
November 9, 2010
2
x
x
Neural Networks
Lecture 16: Counterpropagation
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Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ 1+
+
o
o o
o
3
x
x
November 9, 2010
2
x
x
Neural Networks
Lecture 16: Counterpropagation
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Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ 1+
+
o
o o
o
x
3x
x x
November 9, 2010
2
Neural Networks
Lecture 16: Counterpropagation
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Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ 1+
+
o
o o
o
x
3x
x x
November 9, 2010
2
Neural Networks
Lecture 16: Counterpropagation
25
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ 1+
+
o
o o2
o
x
3x
x x
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
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Counterpropagation – Euclidean Distance
… and so on,
possibly with reduction of the learning rate…
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
27
Counterpropagation – Euclidean Distance
Example of competitive learning with three hidden neurons:
+
+
+ 1+
+
o
o o
o2
x
x
x 3x
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
28
The Counterpropagation Network
After the first phase of the training, each hidden-layer
neuron is associated with a subset of input vectors.
The training process minimized the average angle
difference or Euclidean distance between the weight
vectors and their associated input vectors.
In the second phase of the training, we adjust the
weights in the network’s output layer in such a way
that, for any winning hidden-layer unit, the network’s
output is as close as possible to the desired output for
the winning unit’s associated input vectors.
The idea is that when we later use the network to
compute functions, the output of the winning hiddenlayer unit is 1, and the output of all other hidden-layer
units is 0.
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
29
Counterpropagation – Cosine Similarity
Because there are two output neurons, the weights in the
output layer that receive input from the same hidden-layer unit
can also be described by 2D vectors. These weight vectors are
the only possible output vectors of our network.
network output
if H3 wins
O
( w13O , w23
)
network output
if H1 wins
O
( w11O , w21
)
O
( w12O , w22
)
network output
if H2 wins
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
30
Counterpropagation – Cosine Similarity
For each input vector, the output-layer weights that are
connected to the winning hidden-layer unit are made more
similar to the desired output vector:
O
( w13O , w23
)
O
( w11O , w21
)
O
( w12O , w22
)
O
(w11O,new , w21
, new )
( y1 , y2 )
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
31
Counterpropagation – Cosine Similarity
The training proceeds with decreasing step size , and after its
termination, the weight vectors are in the center of gravity of
their associated output vectors:
O
13
O
23
(w , w )
Output associated with
H1
H2
H3
O
( w12O , w22
)
O
( w11O , w21
)
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
32
Counterpropagation – Euclidean Distance
At the end of the output-layer learning process, the outputs of
the network are at the center of gravity of the desired outputs of
the winner neuron.
o
o 2o
o
x
x
x x3
+ +
+ 1+
+
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
33
The Counterpropagation Network
Notice:
• In the first training phase, if a hidden-layer unit does not win
for a long period of time, its weights should be set to random
values to give that unit a chance to win subsequently.
• It is useful to reduce the learning rates ,  during training.
• There is no need for normalizing the training output
vectors.
• After the training has finished, the network maps the training
inputs onto output vectors that are close to the desired ones.
• The more hidden units, the better the mapping; however, the
generalization ability may decrease.
• Thanks to the competitive neurons in the hidden layer, even
linear neurons can realize nonlinear mappings.
November 9, 2010
Neural Networks
Lecture 16: Counterpropagation
34