1. Simple Learning: Hebbian Learning and the Delta Rule
Psych /719
Feb 1, 2001

2. “Correlational” Learning
Suppose we have a set of samples from some domain we’re 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.

3. 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

4. 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?

5. 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

6. Example: c=a (u=0.25) a b c Dw1 Dw2 a b c w1 w2 -1 -1 -1 .25 .25 1 -1 1 -1 1
Total

7. A Failure: c=a or b a b c Dw1 Dw2 a b c w1 w2 -1 -1 -1 .25 .25 1 -1 1 1 1
Total

8. With Biases… d = a or b (unit c as bias unit)
Dw1
Dw2 c
Dw3 a b d w1 w2 c w3 -1 1 1 1
Total

9. A Real Failure a b e Dw1 Dw2 c Dw3 d Dw4 a b e w1 w2 c w3 d w4 1 1 -1 -1
Total

10. Properties of the Hebb Rule
What if unit outputs are always positive?
What happens over a long period of time?
Doesn’t always work (even if solution exists).
The essential problem: weight value is only a function of behavior of units it connects.

11. Something More Sophisticated: The 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

12. The “Failure” Revisitedx e b c d
Dw1
Dw2
Dw3
Dw4 a b x w1 w2 c w3 d w4 1 1 1 1 -1 -2 -1 -2
Total
And so on….

13. 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.

14. Proof...

15. Limitations
a b
-1 -1
1 -1
-1 1
1 1
a or b -1 1
a and b -1 1 -1 1
a xor b
Cannot solve problems that are not linearly separable (e.g., XOR)
1,1
-1,-1
-1,1
1,-1
I’m going to a Penguins game
on Monday or Wednesday.
Implies that I’m not going to
BOTH games!

16. 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