Spring 2006 Artificial Intelligence COSC40503 week 5

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Spring 2006 Artificial Intelligence
COSC 40503week 5

• Antonio Sanchez
• Texas Christian University

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About Learning and Behavior
Herbert Simon (1916-2001)

• Learning is any change in a system that produces
  a more or less permanent change in its capacity
  for adapting to its environment.
• Human beings, viewed as behaving systems, are
  quite simple. The apparent complexity of our
  behavior over time is largely a reflection of the
  complexity of the environment in which we find
  ourselves.

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Connectionism

• The concept of a Neuron has invited many lines of
  research.
• Yet the power of living neurons lies in both
• Their huge number 1010 in humans, 104 in small
  bug
• Their connectivity 105 in humans
• It is therefore by the power of the
• ganglia (1015) that we present a
• complex and intelligent behavior
• Neurons can be model as digital
• or continuous systems

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A simple Neuron Model

• Orthogonally fh 0
• Normality ff 1 execute fi
  nfi/sqrt(nf.nf)
• Soma ni,j ? ai,kdk,j
• for all k dendrites
• Linear Axon ai,j ni,j
• Non Linear Axon if ni,j gt Thresholdi,j
• then ai,j 1 else
  ai,j 0
• Lineal compensation ?di,j ?ai,kai,j
  0 lt ? lt 1
• k is entry axon
  j exit axon
• N pattern compensation ?di,j
  ??i,k,hai,j,h for all h patterns

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A LinearNeuronal Network
It can learn a linear function such as
Exclusive OR
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A Hidden Layer Neuronal Network
Exclusive OR
Th 1
Th 1
Th1
Excel file
Source Rumelhart, D. E., Hinton, G. E.,
Williams, R. J. (1986a). Learning internal
representations by error propagation. In D. E.
Rumelhart, J. L. McClelland (Eds.), Parallel
distributed processing Explorations in the
microstructure of cognition. Vol. 1 Foundations
(pp. 318--362). Cambridge, MA MIT Press.
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Credit Assignment using Gradient Descent method
to change the dendritic weights

• Change the function for firing neurons from a
  step function to a continous one with a close
  behavior, e.g. a sigmoid axon
• a(j) 1/(1e-n(j))
• Calculate the derivative of the sigmoid
• da(j)/dn(j) a(j)(1-a(j))
• Determine output error
• e(output) TrueValue - a(j)
• Determine the axon error
• ea(j) a(j)(1-a(j))e(output)
• Minimize E by gradient descent change each
  weight by an amount proportional to the partial
  derivative
• ?d aea(j)a(h)

If slope is negative ? increase n(j) If slope is
positive ? decrease n(j) Local minima for Error
are the places where derivative equals zero
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Faster Convergence Momentum rule

• Add a fraction (amomentum) of the last change to
  the current change

?d(t) aea(j)a(h)ß?d(t-1)
Study and Run Excel file
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Extended Hidden LayerNeuronal Network
Hidden Layer
Inputs
Outputs
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A Lesson from Mother NatureUsing the
Scientific Method Observation
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Hypothesis
When an axon of cell A is near enough to excite
cell B and repeatedly or persistently takes part
in firing it, some grow process or metabolic
change takes place in one or both cells such that
As efficiency, as one of the cells firing B, is
increased Donald O. Hebb 1949
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Synthesis and Validation
The connectivity of a brain is product of its
interaction with its environment
High Interaction
Low Interaction
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Argumentation

• In any case the connectivity is due to at least
  the following
• three factors
• Time
• Born, Child, Young, Adult
• Interaction
• Low, Medium, High
• Activity of the brain
• Low, Medium, High

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How about storing Information
2 bit net for
Th 1
Value -1
Th 0
Value -1
Th 1
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How about storing Knowledge
2 bit net for gt
Th 1
Value 1
Th 1
Value 1
Value -1
Th 0
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Digital Neural Network requirements

• Extrapolating the previous data to Gigabytes of
  knowledge and
• information, we would require an enormous amount
  of cells
• and connections, some like 1011 for a full
  Gigabyte of
• information and knowledge
• However we must take into account three important
  facts for the
• case of natural neurons
• They are not binary digital, but continuous
• There is a lot of implicit coding in the ganglia
• They do not store all the information, but only
  important patterns of them (aka knowledge)

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Some important knowledge on Artificial Neural
Networks

• They are slow but once they learn they perform
  beautifully
• Treat the threshold as negative dendrite with an
  axon with value of 1
• To speed up convergence try with random seeds for
  the initial weights
• If they do not converge use another set of random
  weights
• To reduce the size of the network, use binary
  coding for both the inputs and the outputs
• Use other deterministic filters you deem
  necessary such as the mean, sd
• Do not overload the network with more patterns
  than a maximum of 20 to 25 of the number of
  neurons
• A single hidden layer is enough, it should
  comprised about half of the input cells and/or
  twice the number output cells

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Linear Recurrent Networks

• Associative Memory
• Hopfield Network

Excel files
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Unsupervised Learning
So far we have talked about training with a
purpose, I.e. showing some examples and give some
type of compensation, this is called supervised
learning. Yet following Donald O. Hebb
hypothesis, we do not event need to supervised
the learning process, this will take place
anyway!

• Examples
• Kohonen Maps
• Bayesian Nets

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Bayesian Belief Networks
Based on the joint probability concepts of Thomas
Bayes, these networks are used to obtain
inference based on the connection that occurs
within related events, here the basic equations
(try to remember them) p(A ? B) p(A) p(B) -
p(A n B) p(A n B) p(A)p(BA) p(B)p(AB)
p(A,B) p(A n B) 1 - p(A ? B) If p(A n B)
p(A)p(B) then A B are independent events and
no inference can be obtained from having one
event happening If p(A n B) 0 then A B are
mutually exclusive events, that is one cannot
happen if the other is present
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Bayesian Belief Networks 2
An important extension is that p(A,B,C)
p(A)p(BA)p(CA,B) p(B)p(AB)p(CA,B)
p(A)p(CA)p(BA,C) p(B)p(CB)p(AC,B)
p(C )p(AC)p(BA,C) p(A n B n C) For the
general case p(x1,,xn) p p(xi E) for all
i where E is the required a priori evidence
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Bayesian Belief Networks 3
The key aspect is to arrange the events in such a
fashion that we obtain an adequate tree of the
joint probabilities for various events, such that
we can assume that p(F,P,C)
p(C)p(PC)p(FC,P) is computed only as
p(C) p(PC) p(F P)
And
p(E,P,C,S,F) p(C)p(PC)p(SC)p(EC,P,S)p(
FC,P,S,E) is computed only as
p(C)p(SC)p(PC)p(EP,S)p(FP)
See Excel files