Simple Learning: Hebbian Learning and the Delta Rule

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Simple LearningHebbian Learning andthe Delta
Rule

• Psych 85-419/719
• Feb 1, 2001

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Correlational Learning

• Suppose we have a set of samples from some domain
  were interested in.
• People in this class age, height, weight, etc.
• The set of variables (e.g., age, weight) is a
  vector of numbers (22, 30, 19), (175, 130, 200)
• We can compute the correlation between any of
  those vectors. Can predict one variable from
  another.

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In Statistics...
We gather samples
250
We can use a regression to predict one variable
from another
weight
For novel instances, use regression line to
guess what predicted variable is.
100
48
84
height
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Learning

• Decomposability of the problem treat each output
  unit as a simpler problem to solve

So how do we update the weights from input units
to outputs?
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The Hebb Rule

• The change in weight from unit i to j is the
  product of the two units activities, and a
  scaling factor u
• If both units activities are positive, or both
  negative, weight goes up
• If signs are opposite, weight goes down

j
wi,j
i
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Example ca (u0.25)
-1 -1
-1
.25 .25
.25 -.25
1
1 -1
-1
.25 -.25
-1 1
1
.25 .25
1 1
1.0 0.0
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A Failure ca or b
-1 -1
-1
.25 .25
.25 -.25
1
1 -1
-.25 .25
1
-1 1
1
.25 .25
1 1
0.75 0.75
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With Biases d a or b (unit c as bias unit)
-1 -1 1
-1
.25 .25 -.25
.25 -.25 .25
1
1 -1 1
-.25 .25 .25
1
-1 1 1
1
.25 .25 .25
1 1 1
0.75 0.75 0.75
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A Real Failure
1 -1 1 -1
1
.25 -.25 .25 -.25
1
.25 .25 .25 .25
1 1 1 1
-1
-.25 -.25 -.25 .25
1 1 1 -1
-1
-.25 .25 .25 -.25
1 -1 -1 1
0.0 0.0 0.75 0.0
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Properties of the Hebb Rule

• What if unit outputs are always positive?
• What happens over a long period of time?
• Doesnt always work (even if solution exists).
• The essential problem weight value is only a
  function of behavior of units it connects.

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Something More SophisticatedThe Delta Rule

• Weight change a function of activity in the from
  unit (unit j), and the error, e on the to unit i.
• This means weight updates are implicitly a
  function of the overall network behavior

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The Failure Revisited
e
x
Dw4
Dw1
Dw2
Dw3
a
c
b
d
1
1 -1 1 -1
1
.25 -.25 .25 -.25
1
1
.25 .25 .25 .25
1 1 1 1
-2
-1
-.5 -.5 -.5 .5
1 1 1 -1
-2
-1
-.5 .5 .5 -.5
1 -1 -1 1
-.5 0.0 0.5 0.0
And so on.
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Properties of the Delta Rule

• Can be proven that it will converge to the weight
  vector that produces the global minimum possible
  sum squared error between targets and outputs.

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Proof...
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Limitations

• Cannot solve problems that are not linearly
  separable (e.g., XOR)

Im going to a Penguins game on Monday or
Wednesday.
Implies that Im not going to BOTH games!
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Next Class Pattern Association

• Read handout
• And Chapter 11 of PDP1
• Optional read Chapter 9 of PDP1 (a brush up on
  linear algebra)
• Homework 2 handed out