Transcript NNets15_CompetitiveLearning

Unsupervised learning: simple competitive
learning
Biological background:
Neurons are wired topographically, nearby neurons connect to nearby
neurons. In visual cortex, neurons are organized in functional
columns.
Ocular dominance columns: one region responds to one eye input
Orientation columns: one region responds to one direction
Self-organization as a principle of neural development
Competitive learning
--finite resources: outputs ‘compete’ to see which will win via
inhibitory connections between them
--aim is to automatically discover statistically salient features of
pattern vectors in training data set: feature detectors
--can find clusters in training data pattern space which can be used
to classify new patterns
--basic structure:
Outputs
inputs
input layer fully connected to output layer
input to output layer connection feedforward
output layer compares activation's of units following presentation
of pattern vector x via (sometimes virtual) inhibitory lateral
connections
winner selected based on largest activation
winner- takes-all (WTA)
linear or binary activation functions of output units.
Very different from previous (supervised) learning where we pay
our attention to input-output relationship, here we will look at the
pattern of connections (weights)
More formally, if we have a d dimensiona l input vect or x
and M outputs yi we have :
d
yi   w ji x j
j 1
where :
d
wi   w ji  1
2
j 1
for all i  1, ..., M. We say that neuron i * is the winner if :
yi *  yi
for all i, i  i *
and could make this a binary function by defining :
1 if
vj  

y j  yi
for all i, i  j
0 else
Simple competitive learning algorithm
Initialise all weights to random values and normalise (so that || w ||=1)
loop until stopping criteria satisfied .
choose pattern vector x from training set
Compute distance between pattern and weight vectors
|| Xi - W ||
find output unit with largest activation ie ‘winner’
i* with the property that
|| Xi* - W ||< || X i - W ||
update the weight vector of winning unit only with
W(t+1) = W(t)+h (t) (Xi - W (t))
end loop
NB choosing the largest output is the same as choosing the
vector w that is nearest to x since:
a) w.x = wTx = ||w|| ||x|| cos(angle between x and w)
= ||w|| ||x|| if the angle is 0
b) ||w - x||2 = (w1-x1) 2 + (w2-x2) 2
= w1 2 + w2 2 + x1 2 + x2 2 – 2(x1w1 + x2w2)
= ||w|| 2 + ||x|| 2 - 2 wTx
Since ||w|| = 1 and ||x|| is fixed, minimising ||w - x||2 is
equivalent to maximising wTx
Therefore, as we only really want the angle, WLOG only
consider inputs with ||x|| =1
EG h = 0.5
W1 = (-1, 0)
W2 = (0,1)
X = (1, 0)
Y1= -1, y2 = 0 so y2 wins
W2-> (0.5, 0.5)
Y1= -1, y2 = 0.5 so y2 wins
W2-> (0.75, 0.25)
Y1= -1, y2 = 0.75 so y2 wins
W2-> (0.875, 0.125)
Etc etc
How does competitive learning work?
can view the points of all input vectors as in contact with surface of
hypersphere (in 2D: a circle)
distance between points on surface of hypersphere =
degree of similarity between patterns
Outputs
Initial state
inputs
Outputs
inputs
with respect to the incoming signal, the weight of the yellow line is
updated so that the vector is rotated towards the incoming signal
W(t+1) = W(t)+h (t) (Xi - W (t))
Thus the weight vector becomes more and more similar to the input
ie a feature detector for that input
When there are more input patterns, can see each weight vector
migrates from initial position (determined randomly) to centre of
gravity of a cluster of the input vectors.
Thus: Discover Clusters
Outputs
inputs
Eg on-line k-means. Nearest centre updated by:
centrei(n+1) = centrei + h (t) (Xi - centrei(t))
Error minimization viewpoint: Consider error minimization
across all patterns N in training set. Aim is to decrease error E
E = S ||Xi - W (t)||
2
For winning unit k when pattern is Xi the direction the weights
need to change in a direction (so as to perform gradient descent)
determined by (from previous lectures):
W(t+1) = W(t)+h (t) (Xi - W (t))
Which is the update rule for supervised learning (remembering
that in supervised learning, W is O, the output of the neurons)
ie replace W by O and we recover the adaline/simple gradient
descent learning rule
Enforcing fairer competition
Initial position of weight vector of an output unit may be in region
with few, if any, patterns (cf problems of k-means)
Many never or rarely become a winner and so weight vector may not
be updated preventing it finding richer part of pattern space
DEAD UNIT
Or, initial position of a weight vector may be close to a large
number of patterns while most other unit’s weights are more distant
CONTINUAL WINNER, but weights will change little over time
and prevent other units competing
More efficient to ensure a fairer competition where each unit has an
equal chance of representing some part of training data
Leaky learning
modify weights of both winning and losing units but at different
learning rates
hw ( x  w(t ))
w(t  1)  w(t )  
hL ( x  w(t ))
where h w (t) >> h L (t)
has the effect of slowly moving losing units towards denser regions
pattern space.
Many other ways as we will discuss later on
Vector Quantization:
Application of competitive learning
Idea: Categorize a given set of input vectors into M classes using
competitive learning algorithms, and then represent any vector just
by the class into which it falls
Important use of competitive learning (esp. in data compressing)
divides entire pattern space into a number of separate subspaces
set of M units represent set of prototype vectors: CODEBOOK (cf kmeans)
new pattern x is assigned to a class based on its closeness to a
prototype vector using Euclidean distances LABELED
(cf k-nearest neighbours, kernel density estimation)
Example (VQ)
See:
http://www.neuroinformatik.ruhr-uni-bochum.de/ini/VDM/research/gsn/DemoGNG/GNG.html
Topographic maps
Extend the ideas of competitive learning to incorporate the
neighborhood around inputs and neurons
We want a nonlinear transformation of input pattern space onto
output feature space which preserves neighbourhood relationship
between the inputs -- a feature map where nearby neurons respond
to similar inputs
Eg Place cells, orientation columns, somatosensory cells etc
Idea is that neurons selectively tune to particular input patterns in
such a way that the neurons become ordered with respect to each
other so that a meaningful coordinate system for different input
features is created
Known as a Topographic map: spatial locations are indicative of
the intrinsic statistical features of the input patterns: ie close in the
input => close in the output
EG: When the yellow input is active, the yellow neuron is the
winner so when the orange input input is active, we want the
orange neuron to be the winner
Eg Activity-based self-organization (von der Malsburg, 1973)
incorporation of competitive and
cooperative mechanisms to generate
feature maps using unsupervised learning
networks
Biologically motivated: how can
activity-based learning using highly
interconnected circuits lead to orderly
mapping of visual stimulus space onto
cortical surface? (visual-tectum map)
cortical units
visual space
2 layer network each cortical unit fully connect to visual space via
Hebbian units
Interconnections of cortical units described by ‘Mexican-hat’ function
(Garbor function): short-range excitation and long-range inhibition
After learning (see original paper for details), a topographic map
appears.
However, input dimension is the same as output dimension
Kohonen simplified this model and called it
Kohonen’s self-organizing map (SOM) algorithm
More general as it can perform dimensionality reduction
SOM can be viewed as a vector quantisation type algorithm
Kohonen’s self-organizing map (SOM) Algorithm
set time, t=0
initialise learning rate h (0)
initialise size of neighbourhood function
initialize all weights to small random values
inputs
Loop until stopping criteria satisfied
Choose pattern vector x from training set
Compute distance between pattern and weight vectors for each
output unit
|| x - Wi(t) ||
Find winning unit from minimum distance
i*: || x - Wi*(t) || = min || x - Wi(t) ||
Update weights of winning and neighbouring units using
neighbourhood functions
wij(t+1)=wij(t)+ h (t) h(i,i*,t) [xj- wij(t)]
note that when h(i,i*)=1 if i=i* and 0 otherwise, we have
the simple competitive learning
Decrease size of neighbourhood
when t is large we have h(i,i*,t)=1 if i=i* and 0 otherwise
Decrease learning rate h (t)
when t is large we have h
(t) ~ 0
Increment time t=t+1
end loop
Generally, need a LARGE number of iterations
Neighbourhood function
Relates degree of weight update to distance from winning unit, i* to
other units in lattice. Typically a Gaussian function
 || i  i* ||2 

h(i, i*, t )  exp  
2 
 2 (t ) 
When i=i* , distance is zero so h=1
Note h decreases monotonically with distance from winning unit and
is symmetrical about winning unit
Important that the size of neighbourhood (width) decreases over time
to stabilize mapping, otherwise would get noisy version of
competitive learning
Biologically, development (self-organization) is governed by
the gradient of a chemical element, which generally preserves
the topological relation.
Kohonen (and others) has shown that the neighborhood function
could be implemented by a difusing neuromodulatory gas such
as nitric oxide (which has been shown to be necessary for
learning esp. spatial learning)
Thus there is some biological motivation for the SOM
Weight Adaptation
two phases during training: ordering and convergence
Ordering Phase
-Topographic ordering of weight vectors of output units to untangle
map so that weight vectors are evenly spread
Example:
W
1
x
W3
W2
W1 is closest to input x
unit 1 is winner
W3 is closer to x than W2
W
W2 moves closer to x
since h(1,2) > 0;
W3 does not move as h(1,3) = 0
W3
1
W2
W
1
W2
After x is shown many times
W2 moves closer to W1
W3
Finally, the weights will have
the correct order
Completion of ordering phase
Convergence phase
Topographic map adapts to probability density of input patterns
in training data
However, it is important that ordering is maintained
Thus learning rate should be small to prevent large movements of
weight vector but non-zero to avoid pathological states (getting
caught in odd minima etc)
Neighbourhood size should also start off relatively small
containing only close neighbours and eventually shrinking to
nearest neighbours (or less)
Problems
SOM tends to overrepresent regions of low density and
underrepresent regions of high density
Wx  p (x)
2/3
where p(x) is the distribution density of input data
Also, can have problems when there is no natural 2d ordering
of the data.
Visualizing network training
Consider simple 2D lattice of units
1
2
3
4
5
6
Six output units in 2 D space: set up neighbourhood function
according to some two dimensional structure, connect units which
are close together with a line
1
3
4
5
6
Initial random distribution of
weight vectors of each unit in
2D
2
1
3
Ordered weight vectors after
training
2
4
5
6
Example 2D to 2D
Example (JAVA)
See:
http://www.neuroinformatik.ruhr-unibochum.de/ini/VDM/research/gsn/DemoGNG/GNG.html
Semantic Maps
In the previous demos etc, we only viewed the feature map by
visualising the weight vectors of the neurons.
Another method of visualisation is to view class labels assigned to
neurons in a 2d lattice depending on how they respond to (unseen)
test patterns: contextual or semantic maps
In this way the SOM creates coherent regions by grouping sets of
labels with similar features together
Useful tool in data mining and for visualising complex high
dimensional data
Example: Animal and attributes, 16 animals with 13 attributes
Animal
small
is medium
big
2 legs
4 legs
has hair
hooves
mane
feathers
hunt
likes run
to fly
swim
1 1 1 1 1
0 0 0 0 0
1 0 0 0 0 1 0 0
0 1 1 1 1 0 0 0
see Haykin’s book
0 0
0 0
0
0
output
10x10 neurons
29 D inputs
lion=(0,0,0,0,0,0,0,0,0, 0.2, 0,0,0, 0,0,1,0,1,1,0,1,0,1,1,0,0)
0,0,0,0,0,0,0,0,0, 0.2, 0,0,0 =animal code
Train for 2000 interations of all animals (until convergence)
Next the code for each animal is presented but with the animal code
all zeroes. We then assign the neuron to the animal for which it
gives the largest response