The Hebbian Learning Rule

Simon Dennis and Rachael Gibson
Special thanks to Jill White

Table of Contents

Vector Exercises

 In the following exercises, you will build a vector network that learns to discriminate between stored items and new items (see figure 1).

Figure 1: A vector memory network.

Follow the instructions below to create the network and then work through the exercises.

  1. Load BrainWave
  2. Select 'New Hebbian Network'
  3. Set up the vector memory network:
  4. Create the data sets: Exercise 1: Train the network for one epoch and record the weights in the TRAIN FROG row of the following table. How have the weights changed?

    Weight 1Weight 2
    TRAIN FROG  
    TRAIN FROG & KOALA  
    TRAIN FROG & TOAD  

    Exercise 2: Test each of the items, FROG, TOAD and KOALA, and record the match values (the activation of the output unit) in the second table. Explain the match values.

    TEST FROGTEST TOADTEST KOALA
    TRAIN FROG   
    TRAIN FROG & KOALA   
    TRAIN FROG & TOAD   

    Exercise 3: Train the network for one more epoch and test again. What happens to the match values after a second training trial? Why?

    Exercise 4: Add KOALA to the input set and an output value of 1 to the output set. Zero the weights (using the ACTIONS menu) and retrain the network on the updated input set. Test the network as before, recording the values in the table in the TRAIN FROG & KOALA row.

    Exercise 5: Delete KOALA from the input set and add TOAD. Zero the weights, retrain and test as above, recording the values in the TRAIN FROG & TOAD row. You should have six weight values and nine match values for each training trial. Create a graph of the match values after the first training trial: plot three lines, one for each test item, against the three training conditions. Explain the shape of each line on the graph.

    Exercise 6: For each of the three training conditions (FROG alone, FROG & KOALA, FROG & TOAD):

    1. Draw the geometric (graphical) representation of the weights,
    2. Provide the algebraic representation of the weights.

    Matrix Exercises

    In the following exercise you will use a matrix memory network to store and recall pairs of items.

    Exercise 7: Load the simulator, BrainWave. From the NETWORKS menu - select Matrix Model (1). What rank tensor does this network implement? What are its dimensions?

    Exercise 8: The items in this exercise are:

    The input set contains the items FROG, KOALA and SNAIL, paired with items in the output set FLIES, LEAVES and LETTUCE, respectively. Another input item, TOAD [0.5, 0.4, 0.6, 0.45], can be used to test the network on unfamiliar input. Calculate the similarity value (i.e. dot product) of the items FROG, KOALA, SNAIL and TOAD with themselves, and each other, and record the values in the table below:

    FROGKOALASNAILTOAD
    FROG    
    KOALA    
    SNAIL    
    TOAD    

    Exercise 9: Train the network for one epoch. Test each of the items FROG, KOALA, SNAIL and TOAD. What output is produced in each case? (Give the output pattern and also describe the output patterns in terms of their similarity to FLIES, LEAVES and LETTUCE).

    FROG   
    KOALA   
    SNAIL   
    TOAD   

    Exercise 10: Give the algebraic equation that describes the matrix memory formed from the three pairs of associates:

    Exercise 11: Give the equations that describe each of the retrievals in exercise 9. Use the similarity measures from the table above to simplify each equation to a weighted sum of the target patterns.
      FROG
      KOALA
      SNAIL
      TOAD

    Tensor Exercises

    In these exercises, you will use both rank two and three tensor networks to store and recall triples of items.

    Exercise 12: Load the simulator, BrainWave. From the NETWORKS menu - Matrix Model (2). What rank tensor does this network implement?







    Exercise 13: The items in this exercise are:

    Cues:

    Relations:
      EATS [0.5, -0.5, -0.5, 0.5]
      LIVES-IN [0.5, 0.5, -0.5, -0.5]
    Targets:
      FLIES [0.7, 0.5, 0.5]
      LEAVES [0.7, -0.5, -0.5]
      LETTUCE [0, -0.7, 0.7]
      POND [0.89, 0.43, -0.22]
      TREE [-0.22, 0.76, 0.62]
      SHELL [0.43, -0.5, 0.76]
    Notice that the vectors for the cues are the same as those used above. Also notice that EATS and LIVES-IN are orthogonal to each other - that is they have a dot product of zero.

    Calculate the similarity (dot product) table for the targets.

    FLIESLEAVESLETTUCEPONDTREESHELL
    FLIES      
    LEAVES      
    LETTUCE      
    POND      
    TREE      
    SHELL      

    Exercise 14: The cue+relation input set contains the items FROG-EATS, KOALA-EATS, SNAIL-EATS, FROG-LIVES_IN, KOALA-LIVES_IN and SNAIL-LIVES_IN, paired with items in the output set FLIES, LEAVES, LETTUCE, POND, TREE, and SHELL, respectively. Two other input items, TOAD-EATS and TOAD-LIVES_IN, can be used to test the network's response to unfamiliar input.

    Train the network for one epoch. Test each of the items FROG-EATS, KOALA-EATS, SNAIL-EATS, FROG-LIVES_IN, KOALA-LIVES_IN, SNAIL-LIVES_IN, TOAD-EATS and TOAD-LIVES_IN. What output is produced in each case? (Give the output pattern and also describe the output patterns in terms of their similarity to FLIES, LEAVES, LETTUCE, POND, TREE and SHELL)

    Exercise 15: How does the performance of this network compare with the performance of the network in Exercise 8? Why is it not as good?

    Exercise 16: Give the algebraic equation that describes the matrix memory formed from the three pairs of associates:

    Exercise 17: Give the equations that describe each of the retrievals from exercise 14. Use the similarity measures from the table above to simplify each equation to a weighted sum of the target patterns.

    Exercise 18: From the NETWORKS menu - select Matrix Model (3). What rank tensor does this network implement?



    Exercise 19: The inputs and outputs for this network are the same as for the previous one, but the connections and hidden SigmaPi units perform different calculations on the inputs to try and achieve the correct outputs. Train the network for one epoch. Test each of the items FROG-EATS, KOALA-EATS, SNAIL-EATS, FROG-LIVES_IN, KOALA-LIVES_IN, SNAIL-LIVES_IN, TOAD-EATS and TOAD-LIVES_IN.

    What output is produced in each case? (Give the output pattern and also describe the output patterns in terms of their similarity to FLIES, LEAVES, LETTUCE, POND, TREE and SHELL).

    Exercise 20: Which of the two networks performs the memory task better? Why?

    Exercise 21: Give the algebraic equation that describes the matrix memory formed from the three pairs of associates:

    Exercise 22: Give the equations that describe each of the cued recall tests from question 19. Use the similarity measures from the table above to simplify each equation to a weighted sum of the target patterns.
      FROG-EATS

      KOALA-EATS

      SNAIL-EATS

      FROG-LIVES_IN

      KOALA-LIVES_IN

      SNAIL-LIVES_IN

      TOAD-EATS

      TOAD-LIVES_IN


    In this section, we have been looking at the way in which tensors of rank one, two and three can be used to store information.

    Objective Checklist

    The following is a check list of skills and knowledge which you should obtain while working on this chapter. Go through the list and tick off those things you are confident you can do. For any item outstanding, you should refer back to the appropriate section or consult your tutor.